C_{c}^{\infty}(\mathbb{R}^{d})$ such that $\phi=1$ on a neighborhood of
$[0,nL]^{d}$,

$\displaystyle\mathcal{Z}_{Z}^{[0,nL]^{d}}(u,u)=\langle\phi
Z,\mathbbm{1}_{[0,nL]^{d}}u^{2}\rangle$
$\displaystyle=\sum_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}\langle\phi
Z,\mathbbm{1}_{[0,nL]^{d}}\eta_{k}^{2}u^{2}\rangle$
$\displaystyle=\sum_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}\mathcal{Z}_{Z}^{kL+[-l,L+l]^{d}}(\eta_{k}u,\eta_{k}u).$

Therefore, by Lemma 4.29,

$\mathcal{E}_{W,\mathcal{Z}}^{[0,nL]^{d}}(u,u)\geq\sum_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}\Big{\\{}\mathcal{E}_{W,\mathcal{Z}}^{kL+[-l,L+l]^{d}}(\eta_{k}u,\eta_{k}u)-\frac{K}{l^{2}}\lVert\eta_{k}u\rVert_{L^{2}(kL+[-l,L+l]^{d})}^{2}\Big{\\}}.$

and thus
$\mathcal{E}_{W,\mathcal{Z}}^{[0,nL]^{d}}\succ\bigoplus_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}[\mathcal{E}_{W,\mathcal{Z}}^{kL+[-l,L+l]^{d}}-\frac{K}{l^{2}}\mathfrak{I}]$
(where $\mathfrak{I}$ is as in Definition 4.26), from which we conclude the
estimates (use the discussion in Definition 4.26).

(b) As given in [23, Proposition 4.3], there exist smooth functions
$\alpha_{1}$ and $\alpha_{2}$ on $\mathbb{R}^{d}$ and a $K>0$ (only depending
on $d$) such that

$\displaystyle\operatorname{supp}(\alpha_{1})\subseteq U_{L}\setminus
B(\partial U_{L},\tfrac{R}{2}),\quad\operatorname{supp}(\alpha_{2})\subseteq
B(\partial U_{L},R),$ $\displaystyle\alpha_{1}^{2}+\alpha_{2}^{2}=1\mbox{ on a
neighborhood of
}U_{L},\qquad\Big{\|}\sum_{j=1}^{2}\lvert\nabla\alpha_{j}\rvert^{2}\Big{\|}_{L^{\infty}(\mathbb{R}^{d})}\leq
KR^{-2}.$

Recall the definitions of $U_{L}^{R}$ and $H^{1}_{\mathfrak{m},R}(U^{R}_{L})$
from Definition 4.20. The map

$\Phi:L^{2}(U_{L})\rightarrow L^{2}(U_{L})\oplus L^{2}(U^{R}_{L}),\quad
u\mapsto\alpha_{1}u\oplus\alpha_{2}u$

is an isometry and $\Phi(e^{W}H^{1}(U_{L}))\subseteq
e^{W}H^{1}_{0}(U_{L})\oplus e^{W}H^{1}_{\mathfrak{m},R}(U^{R}_{L})$.

Observe that
$\mathcal{Z}^{U_{L}}(\alpha_{2}u,\alpha_{2}u)=\mathcal{Z}^{U_{L}^{R}}(\alpha_{2}u,\alpha_{2}u)$
as $\operatorname{supp}\alpha_{2}\cap U_{L}\subset U_{L}^{R}$. Therefore, by
Lemma 4.29, for $u\in H^{1}(U_{L})$,

$\mathcal{E}^{U_{L}}(u,u)\geq\mathcal{E}^{U_{L}}(\alpha_{1}u,\alpha_{1}u)+\mathcal{E}^{U_{L}^{R}}(\alpha_{2}u,\alpha_{2}u)-\sum_{j=1}^{2}KR^{-2}\lVert\alpha_{j}u\rVert_{L^{2}(U_{L})}^{2}.$

By applying Lemma 4.11 with
$\delta=\frac{e^{-2\|W\|_{L^{\infty}(U_{L})}}}{2}$, we obtain

$\displaystyle\mathcal{E}^{U_{L}^{R}}(\alpha_{2}u,\alpha_{2}u)$
$\displaystyle\geq\frac{e^{-2\lVert
W\rVert_{L^{\infty}(U_{L})}}}{2}\int_{U^{R}_{L}}\lvert\nabla(\alpha_{2}u^{\flat})\rvert^{2}-A\lVert\alpha_{2}u\rVert^{2}_{L^{2}(U^{R}_{L})},$

where

$A\vcentcolon=\frac{e^{-2\|W\|_{L^{\infty}(U_{L})}}}{2}+2^{\frac{s}{1-s}}e^{\frac{2s}{1-s}\lVert
W\rVert_{L^{\infty}(U_{L})}}C_{\operatorname{IP}}^{U_{L}}[H^{s}]^{\frac{2}{1-s}}\llbracket\mathcal{Z}\rrbracket_{H^{s}(U)}^{\frac{1}{1-s}}.$

Therefore,
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U_{L}}\succ(\mathcal{E}^{\scaleobj{0.8}{\mathrm{D}},U_{L}}-KR^{-2}\mathfrak{I})+(\mathcal{E}^{\mathfrak{m},R,U_{L}^{R}}-(KR^{-2}+A)\mathfrak{I})$,
where $\mathcal{E}^{\mathfrak{m},R,U_{L}^{R}}$ is the restriction of
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U_{L}^{R}}$ to
$H_{\mathfrak{m},R}(U_{L}^{R})$, and thus, by Lemma 4.25 (the additional
factor $e^{2\lVert W\rVert_{L^{\infty}(U_{L})}}$ is explained similarly as in
(26)),

$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U_{L},\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda+KR^{-2})+\bm{N}^{\mathfrak{m}}_{0}(U^{R}_{L},2e^{4\lVert
W\rVert_{L^{\infty}(U_{L})}}(\lambda+KR^{-2}+A)).$

By Lemma 4.23 (a), for $R\geq R_{U}$,

$\displaystyle\bm{N}^{\mathfrak{m}}_{0}(U^{R}_{L},2e^{4\lVert
W\rVert_{L^{\infty}(U)}}(\lambda+KR^{-2}+A))$
$\displaystyle\qquad\lesssim_{U}R^{d}L^{d-1}\big{\\{}e^{4\lVert
W\rVert_{L^{\infty}(U)}}(1+\max\\{\lambda,0\\}+KR^{-2}+A)\big{\\}}^{\frac{d}{2}}.$

It remains to apply Lemma A.15, more specifically (67):
$C^{U_{L}}_{\operatorname{IP}}[H^{s}]\lesssim_{s,U}1$. ∎

##  5 The Anderson Hamiltonian with Dirichlet and Neumann boundary conditions

Based on the results obtained in Section 3 and in Section 4, we can give the
definition of the Anderson Hamiltonian $-\Delta-\xi$ with Dirichlet- and with
Neumann boundary conditions, and show that it is the limit of the operators
$-\Delta-\xi_{\varepsilon}+c_{\varepsilon}$, where the constants
$c_{\varepsilon}$ are defined in (88).

The construction of the Dirichlet Anderson Hamiltonian is given in Section
5.1.

For Neumann boundary conditions we have to deal with the additional boundary
term (see the beginning of Section 4). We impose another assumption
(Assumption 5.7) in order to deal with these boundary terms. More precisely,
we are able to handle this by means of Theorem 4.6 (b) by showing that the
terms $\tilde{Y}_{\varepsilon}^{U}$ (as mentioned in the beginning of Section
4) converge. Let us indicate that the 3D white noise does not satisfy the
conditions of Assumption 5.7, though the 2D white noise does. The additional
assumption and the convergence of the terms $\tilde{Y}_{\varepsilon}^{U}$ are
considered in Section 5.2.

The construction of the Neumann Anderson Hamiltonian is given in Section 5.3.
Finally, in Section 5.4 we consider the integrated density of states
associated to the Anderson Hamiltonian.

Let us now introduce the random variable $M$ with values in $\mathbb{N}_{0}$
such that the conditions of the second part of Theorem 3.3 are satisfied. For
that we first observe the following:

######  Lemma 5.1.

Assume 3.10. Let $U$ be a bounded domain, $\sigma\in(0,\infty)$ and
$\delta_{-}\in(0,\delta)$. Then for $L\geq 1$ and $n\in\mathbb{N}_{0}$

$\displaystyle\lVert
W_{n}\rVert_{C^{\delta_{-}}(U_{L})}\lesssim_{U,\delta_{-},\delta,\sigma}L^{\sigma}2^{-(\delta-\delta_{-})n}\lVert
X\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}.$

Consequently, almost surely $\lim_{n\rightarrow\infty}\lVert
W_{n}\rVert_{\mathcal{C}^{\delta_{-}}(U)}=0$.

###### Proof.

This follows by Lemma A.5 and Corollary A.10, because $W_{n}=G_{n}*X$. The
consequence follows because
$X\in\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})$ almost surely. ∎

###  5.1 The Dirichlet Anderson Hamiltonian

######  Assumptions for this section 5.2.

In this section we assume 3.10.

######  Definition 5.3.

Let $U$ be a bounded domain and let $r\in(1-\delta,1)$. Using the notation of
Theorem 4.6, for $N\in\mathbb{N}_{0}$ we define the following symmetric forms
on $H_{0}^{r}(U)$:

$\mathcal{Z}_{N}[U]\vcentcolon=\mathcal{Z}^{U}_{Y_{N}},\quad\mathcal{Z}^{\varepsilon}_{N}[U]\vcentcolon=\mathcal{Z}^{U}_{Y^{\varepsilon}_{N}}.$
(30)

For $\delta_{-}\in(0,\delta]$ and $\gamma\in(0,\infty)$, we set

$M(U,\delta_{-};\gamma)\vcentcolon=\inf\\{N\in\mathbb{N}\nonscript\>|\nonscript\>\mathopen{}\lVert
W_{n}\rVert_{\mathcal{C}^{\delta_{-}}(U)}\leq\gamma\mbox{ for all }n\geq
N\\}.$ (31)

Recalling the notation from Definition 4.9, for $M=M(U,\delta;1)$ (which
attains its values in $\mathbb{N}_{0}$ by Lemma 5.1), we set

$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}\vcentcolon=\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}\vcentcolon=\mathcal{H}_{W_{M},\mathcal{Z}_{M}[U]}^{\scaleobj{0.8}{\mathrm{D}},U},\quad\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}\vcentcolon=\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}},U}\vcentcolon=\mathcal{H}_{W^{\varepsilon}_{M},\mathcal{Z}^{\varepsilon}_{M}[U]}^{\scaleobj{0.8}{\mathrm{D}},U}.$

Recalling Proposition 4.13, we set

$\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U)\vcentcolon=\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U;W_{M},\mathcal{Z}_{M}[U]),\quad\lambda_{k;\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}(U)\vcentcolon=\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U;W^{\varepsilon}_{M},\mathcal{Z}^{\varepsilon}_{M}[U]).$

######  Theorem 5.4.

For $\varepsilon\in(0,1)$,

$\displaystyle\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}=-\Delta-\xi_{\varepsilon}+c_{\varepsilon}.$
(32)

Let $(\varepsilon_{n})_{n\in\mathbb{N}}$ be a sequence in $(0,1)$ such that
$\varepsilon_{n}\to 0$. Then, there exist a subsequence
$(\varepsilon_{n_{m}})_{m=1}^{\infty}$ and a subset
$\Omega_{1}\subseteq\Omega$ of $\mathbb{P}$-probability $1$ such that on
$\Omega_{1}$ the following holds: for any bounded domain $U$, one has

$\displaystyle\mathcal{H}_{\varepsilon_{n_{m}}}^{\scaleobj{0.8}{\mathrm{D}},U}\xrightarrow{\textnormal{NR}}_{m\rightarrow\infty}\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U},$
(33)

for all $k\in\mathbb{N}$

$\lim_{m\to\infty}\lambda_{k;\varepsilon_{n_{m}}}^{\scaleobj{0.8}{\mathrm{D}}}(U)=\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U),$

and there exist an eigenfunction $\phi_{k}$ of
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}$ corresponding to
$\lambda_{k;\varepsilon_{n_{m}}}^{\scaleobj{0.8}{\mathrm{D}}}$ and
eigenfunctions $\phi_{k,m}$ of
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}_{\varepsilon_{n_{m}}}$
corresponding to
$\lambda_{k;\varepsilon_{n_{m}}}^{\scaleobj{0.8}{\mathrm{D}}}(U)$ for all
$m\in\mathbb{N}$ such that

$\displaystyle\phi_{k,m}\xrightarrow{m\to\infty}\phi_{k}\quad\mbox{ in
}L^{2}(U).$

###### Proof.

(32) follows by our choice of $M$ (see (31)) and by Lemma 1.12 (see also Lemma
4.5) and Theorem 3.3.

Let $\sigma\in(0,1)$. By Theorem 3.3, there exist a subsequence
$(\varepsilon_{n_{m}})_{m=1}^{\infty}$ and a subset
$\Omega_{1}\subseteq\Omega$ of $\mathbb{P}$-probability $1$ such that on
$\Omega_{1}$, for every $N\in\mathbb{N}_{0}$,

$\displaystyle\lim_{m\rightarrow\infty}\lVert
X^{\varepsilon_{n_{m}}}-X\rVert_{\mathcal{C}^{\delta,\sigma}(\mathbb{R}^{d})}=0,\qquad\lim_{m\rightarrow\infty}\lVert
Y^{\varepsilon_{n_{m}}}_{N}-Y_{N}\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}.$

Observe that by Lemma A.5 and Corollary A.10, like in the proof of Lemma 5.1,
for all $\varepsilon\in(0,1)$,

$\displaystyle\lVert
W^{\varepsilon}_{M}-W_{M}\rVert_{\mathcal{C}^{\delta}(U)}\lesssim_{U,\delta,\sigma}\lVert
X^{\varepsilon}-X\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})},$

and that by Theorem 4.6 (a), for $r\in(1-\delta,1)$,

$\displaystyle\llbracket\mathcal{Z}^{\varepsilon}_{M}-\mathcal{Z}_{M}\rrbracket_{H^{r}_{0}(U)}\lesssim_{\delta,p,U}\lVert
Y^{\varepsilon}_{M}-Y_{M}\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}.$

Therefore, the claim follows from Theorem 4.17 and Lemma 4.15. ∎

######  Remark 5.5.

Let $\mathfrak{M}$ be a random variable with values in $\mathbb{N}_{0}$ such
that $\mathfrak{M}\geq M(U,\delta;1)$. Then, almost surely,
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}(U)=\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}_{W_{\mathfrak{M}},\mathcal{Z}_{\mathfrak{M}}[U]}(U)$,
because
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}_{W^{\varepsilon}_{\mathfrak{M}},\mathcal{Z}^{\varepsilon}_{\mathfrak{M}}[U]}(U)=-\Delta-\xi_{\varepsilon}+c_{\varepsilon}=\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}(U)$
and similarly as in Theorem 5.4,
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}_{W_{\mathfrak{M}},\mathcal{Z}_{\mathfrak{M}}[U]}(U)$
is the limit (in the sense of (33)) of
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}_{W^{\varepsilon}_{\mathfrak{M}},\mathcal{Z}^{\varepsilon}_{\mathfrak{M}}[U]}(U)$.
We will apply this in Section 5.4 with $\mathfrak{M}=M(U,\delta;\gamma)$ for
$\gamma\in(0,1]$.

As will be needed in Section 5.4, we give estimates on
$M(U,\delta_{-};\gamma)$ and $\mathcal{Z}_{M}$.

######  Lemma 5.6.

Let $U$ be a bounded domain, $0<\delta_{-}<\delta$ and $\sigma\in(0,\infty)$.
Then, there exists a $C=C(U,\delta_{-},\sigma)\in(0,\infty)$ such that for all
$L\geq 1$ and $\gamma\in(0,\infty)$ one has

$M(U_{L},\delta_{-};\gamma)\leq
1+(\delta-\delta_{-})^{-1}\log_{2}\big{(}C\gamma^{-1}L^{\sigma}\lVert
X\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}\big{)}.$ (34)

Moreover, for $r\in(1-\delta_{-},1)$, $\gamma\in(0,\infty)$ and $L\geq 1$, one
has for $M_{L,\delta_{-},\gamma}\vcentcolon=M(U_{L},\delta_{-};\gamma)$

$\llbracket\mathcal{Z}_{M_{L,\delta_{-},\gamma}}[U_{L}]\rrbracket_{H_{0}^{r}(U_{L})}\lesssim_{U,\delta_{-},\delta,\sigma}\gamma^{-(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}L^{(2+\bm{\mathfrak{b}}(\delta-\delta_{-})^{-1})\sigma}\lVert
X\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}^{(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}\bm{\mathfrak{a}}.$
(35)

###### Proof.

(34) is a direct consequence of Lemma 5.1.

(35) follows by Theorem 4.6 (a) (see (15)) since by definition of
$\bm{\mathfrak{b}}$ and $\bm{\mathfrak{a}}$ (see (5)),

$\lVert Y_{N}\rVert_{\mathcal{C}^{-1+\delta_{-},\sigma}(\mathbb{R}^{d})}\leq
2^{\bm{\mathfrak{b}}N}\bm{\mathfrak{a}},$ (36)

and by using (34). ∎

###  5.2 Stochastic terms for the Neumann Anderson Hamiltonian

######  Assumption 5.7 (Assumption II).

We assume $\delta\in(\frac{1}{2},1)$ and that there exists a
$\delta^{\prime}\in(0,1)$ such that, for each $p\in(1,\infty)$, there exist a
constant $C^{\partial}_{p}\in(0,\infty)$ and a map
$\bm{\varepsilon}_{p}^{\partial}:(0,1)^{3}\times\mathbb{R}^{d}\to(0,\infty)$
with the following properties.

  1. (a)

One has

$\sup_{\varepsilon_{1},\varepsilon_{2},\lambda\in(0,1),x\in\mathbb{R}^{d}}\bm{\varepsilon}_{p}^{\partial}(\varepsilon_{1},\varepsilon_{2};\lambda,x)<\infty$
(37)

and for each fixed $\lambda\in(0,1)$ and $x\in\mathbb{R}^{d}$, one has
$\lim_{\varepsilon_{1},\varepsilon_{2}\downarrow
0}\bm{\varepsilon}_{p}^{\partial}(\varepsilon_{1},\varepsilon_{2};\lambda,x)=0$.

  2. (b)

For every $\varepsilon_{1},\varepsilon_{2},\lambda\in(0,1)$, $p\in(1,\infty)$,
bounded Lipschitz domain $U$, $x\in B(U,1)$ and $\phi\in
C^{2}(\mathbb{R}^{d})$ with
$\lVert\phi\rVert_{\mathcal{C}^{2}(\mathbb{R}^{d})}\leq 1$ and
$\operatorname{supp}(\phi)\subseteq B(0,1)$, one has (for $\phi^{\lambda}_{x}$
see Assumption 3.10)

$\mathbb{E}[\lvert\langle\xi_{\varepsilon_{1}},\mathbbm{1}_{U}\phi^{\lambda}_{x}\rangle\rvert^{p}]\leq
C^{\partial}_{p}\lambda^{(-2+\delta+\delta^{\prime})p}$ (38)
$\mathbb{E}[\lvert\langle\xi_{\varepsilon_{1}}-\xi_{\varepsilon_{2}},\mathbbm{1}_{U}\phi^{\lambda}_{x}\rangle\rvert^{p}]\leq\bm{\varepsilon}_{p}^{\partial}(\varepsilon_{1},\varepsilon_{2};\lambda,x)\lambda^{(-2+\delta+\delta^{\prime})p}.$
(39)

Furthermore, for every bounded domain $U$, we assume that, as
$\varepsilon\downarrow 0$, the distributions
$\mathbbm{1}_{U}\xi_{\varepsilon}$ converge to some limit that is independent
of the mollifier $\rho$, in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ in
probability.

######  Remark 5.8.

In the proof of the Lemma 5.9 we will use the following facts, which are
straightforward to check. For functions
$f,g,\phi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ (for which the following
expressions make sense)

$\displaystyle\langle f*\phi,g\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle=\langle f,\phi(-\cdot)*g\rangle_{L^{2}(\mathbb{R}^{d})},$
$\displaystyle\langle f*\phi(\lambda\cdot),g\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle=\lambda^{-2d}\langle
f(\tfrac{1}{\lambda}\cdot)*\phi,g(\tfrac{1}{\lambda}\cdot)\rangle_{L^{2}(\mathbb{R}^{d})},\quad\lambda>0,$
$\displaystyle(f*g)(\cdot-w)$
$\displaystyle=[f(\cdot-w)]*g=f*[g(\cdot-w)],\quad w\in\mathbb{R}^{d},$
$\displaystyle\langle f*\phi(\cdot-w),g\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle=\langle
f(\cdot-w)*\phi,g\rangle_{L^{2}(\mathbb{R}^{d})}=\langle
f*\phi,g(\cdot+w)\rangle_{L^{2}(\mathbb{R}^{d})},\quad w\in\mathbb{R}^{d}.$

######  Lemma 5.9.

Let $\xi$ be a centered Gaussian noise whose covariance is given by

$\displaystyle\mathbb{E}[\langle\xi,\varphi\rangle\langle\xi,\psi\rangle]=\langle\gamma*\varphi,\psi\rangle_{L^{2}(\mathbb{R}^{d})},\quad\varphi,\psi\in
C_{c}^{\infty}(\mathbb{R}^{d}).$

Suppose one has a bound

$\lvert\gamma\rvert\leq f+g$ (40)

where $g\in L^{\infty}(\mathbb{R}^{d})$ and $f$ satisfies $f(\lambda
x)=\lambda^{-\alpha}f(x)$ with $\alpha<3$ for every $\lambda\in(0,\infty)$ and
$x\in\mathbb{R}^{d}$. Furthermore, suppose $f$ is locally integrable. Then,
$\xi$ satisfies Assumption 5.7.

###### Proof.

As $\xi$ is Gaussian, so is for example
$\langle\xi_{\varepsilon_{1}},\mathbbm{1}_{U}\phi_{x}^{\lambda}\rangle_{\mathbb{R}^{d}}$.
As for Gaussian random variables $Z$ one has
$\mathbb{E}{[|Z|^{p}]}=\mathbb{E}{[|Z|^{2}]}^{\frac{p}{2}}\mathbb{E}{[|X|^{p}]}$,
for $X$ a standard normal random variable, it is sufficient to consider $p=2$.

Let $U$ be a bounded Lipschitz domain and $\phi$ be as in Assumption 5.7.
Observe that (similarly to the first equality in Remark 5.8, using that $\rho$
is symmetric)

$\displaystyle\langle\xi_{\varepsilon_{1}}-\xi_{\varepsilon_{2}},\psi\rangle=\langle\xi*\rho_{\varepsilon_{1}}-\xi*\rho_{\varepsilon_{2}},\psi\rangle=\langle\xi*(\rho_{\varepsilon_{1}}-\rho_{\varepsilon_{2}}),\psi\rangle=\langle\xi,(\rho_{\varepsilon_{1}}-\rho_{\varepsilon_{2}})*\psi\rangle.$

Therefore, for any $\lambda\in(0,1)$ and $x\in\mathbb{R}^{d}$,

$\displaystyle\mathbb{E}[\langle\xi_{\varepsilon_{1}}-\xi_{\varepsilon_{2}},\mathbbm{1}_{U}\phi^{\lambda}_{x}\rangle^{2}]$
$\displaystyle=\langle\gamma*(\rho_{\varepsilon_{1}}-\rho_{\varepsilon_{2}})*(\mathbbm{1}_{U}\phi_{x}^{\lambda}),(\rho_{\varepsilon_{1}}-\rho_{\varepsilon_{2}})*(\mathbbm{1}_{U}\phi_{x}^{\lambda})\rangle_{L^{2}(\mathbb{R}^{d})}.$

We set

$\bm{\varepsilon}_{2}^{\partial}(\varepsilon_{1},\varepsilon_{2};\lambda,x)\vcentcolon=\lambda^{\alpha}\left\lvert\langle\gamma*(\rho_{\varepsilon_{1}}-\rho_{\varepsilon_{2}})*(\mathbbm{1}_{U}\phi_{x}^{\lambda}),(\rho_{\varepsilon_{1}}-\rho_{\varepsilon_{2}})*(\mathbbm{1}_{U}\phi_{x}^{\lambda})\rangle_{L^{2}(\mathbb{R}^{d})}\right\rvert$

One has

$\lim_{\varepsilon_{1},\varepsilon_{2}\downarrow
0}\langle\gamma*\rho_{\varepsilon_{1}}*(\mathbbm{1}_{U}\phi_{x}^{\lambda}),\rho_{\varepsilon_{2}}*(\mathbbm{1}_{U}\phi_{x}^{\lambda})\rangle_{L^{2}(\mathbb{R}^{d})}=\langle\gamma\ast(\mathbbm{1}_{U}\phi_{x}^{\lambda}),(\mathbbm{1}_{U}\phi_{x}^{\lambda})\rangle_{L^{2}(\mathbb{R}^{d})},$

and hence $\lim_{\varepsilon_{1},\varepsilon_{2}\downarrow
0}\bm{\varepsilon}_{2}^{\partial}(\varepsilon_{1},\varepsilon_{2};\lambda,x)=0$.
To prove the bound (37) and (38), it suffices to show

$\sup_{\varepsilon\in(0,1),\lambda\in(0,1),x\in\mathbb{R}^{d}}\lambda^{\alpha}\left\lvert\langle\gamma*\rho_{\varepsilon}*(\mathbbm{1}_{U}\phi_{x}^{\lambda}),\rho_{\varepsilon}*(\mathbbm{1}_{U}\phi_{x}^{\lambda})\rangle_{L^{2}(\mathbb{R}^{d})}\right\rvert<\infty.$

Let us write

$\displaystyle U_{x}^{\lambda}=\lambda^{-1}(U-x).$ (41)

By Remark 5.8 and (40),

$\displaystyle\sup_{x\in\mathbb{R}^{d}}\left\lvert\langle\gamma*\rho_{\varepsilon}*(\mathbbm{1}_{U}\phi_{x}^{\lambda}),\rho_{\varepsilon}*(\mathbbm{1}_{U}\phi_{x}^{\lambda})\rangle_{L^{2}(\mathbb{R}^{d})}\right\rvert$
$\displaystyle=\sup_{x\in\mathbb{R}^{d}}\lvert\langle\gamma(\lambda\cdot)\ast(\mathbbm{1}_{U_{x}^{\lambda}}\phi)\ast\rho_{\varepsilon/\lambda},(\mathbbm{1}_{U_{x}^{\lambda}}\phi)\ast\rho_{\varepsilon/\lambda}\rangle\rvert$
$\displaystyle\leq\lambda^{-\alpha}\langle
f\ast\lvert\phi\rvert\ast\lvert\rho_{\varepsilon/\lambda}\rvert,\lvert\phi\rvert\ast\lvert\rho_{\varepsilon/\lambda}\rvert\rangle+\langle
g(\lambda\cdot)\ast\lvert\phi\rvert\ast\lvert\rho_{\varepsilon/\lambda}\rvert,\lvert\phi\rvert\ast\lvert\rho_{\varepsilon/\lambda}\rvert\rangle.$

Since, using Young’s inequality one can bound the second term by

$\lVert
g\rVert_{L^{\infty}(\mathbb{R}^{d})}\lVert\phi\rVert_{L^{1}(\mathbb{R}^{d})}^{2}\lVert\rho\rVert_{L^{1}(\mathbb{R}^{d})}^{2},$

it comes down to showing

$\sup_{\mu\in(0,\infty)}\langle
f\ast\lvert\phi\rvert\ast\lvert\rho_{\mu}\rvert,\lvert\phi\rvert\ast\lvert\rho_{\mu}\rvert\rangle<\infty.$

$\bullet$ Suppose $\mu\leq 1$. Let $\sigma>d$. By the weighted Young’s
inequality, Theorem A.2, one has

$\langle
f\ast\lvert\phi_{U}\rvert\ast\lvert\rho_{\mu}\rvert,\lvert\phi_{U}\rvert\ast\lvert\rho_{\mu}\rvert\rangle\lesssim_{\sigma}\lVert
w_{\sigma}f\rVert_{L^{1}(\mathbb{R}^{d})}\lVert
w_{-\sigma}\phi\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lVert
w_{-\sigma}\rho_{\mu}\rVert_{L^{1}(\mathbb{R}^{d})}^{2}.$

As $\phi$ is a continuous function with compact support, we have $\lVert
w_{-\sigma}\phi\rVert_{L^{2}(\mathbb{R}^{d})}<\infty$. Since $f$ is locally
integrable and satisfies the scaling property,

$\lVert w_{\sigma}f\rVert_{L^{1}(\mathbb{R}^{d})}=\int_{\partial B(0,1)}\lvert
f(x)\rvert\,\mathrm{d}S(x)\int_{0}^{\infty}r^{d-1-\alpha}(1+r^{2})^{-\frac{\sigma}{2}}\,\mathrm{d}r<\infty.$

Then, we observe, as $\mu\leq 1$,

$\lVert
w_{-\sigma}\rho_{\mu}\rVert_{L^{1}(\mathbb{R}^{d})}=\int_{\mathbb{R}^{d}}(1+\mu^{2}\lvert
x\rvert^{2})^{\frac{\sigma}{2}}\lvert\rho\rvert(x)\,\mathrm{d}x\leq\int_{\mathbb{R}^{d}}(1+\lvert
x\rvert^{2})^{\frac{\sigma}{2}}\lvert\rho\rvert(x)\,\mathrm{d}x<\infty.$

$\bullet$ Now suppose $\mu\geq 1$. By change of variables, see Remark 5.8,

$\displaystyle\langle
f\ast\lvert\phi\rvert\ast\lvert\rho_{\mu}\rvert,\lvert\phi\rvert\ast\lvert\rho_{\mu}\rvert\rangle$
$\displaystyle=\mu^{-\alpha}\langle
f\ast\lvert\phi_{\mu^{-1}}\rvert\ast\lvert\rho\rvert,\lvert\phi_{\mu^{-1}}\rvert\ast\lvert\rho\rvert\rangle.$

Therefore, it reduces to the case $\mu\leq 1$.

As Lemma 5.12 below shows, the estimate (39) implies the convergence of
$\mathbbm{1}_{U}\xi_{\varepsilon}$. To see the independence of the mollifier
$\rho$, we note that the limit $\xi^{U}$ is a centered Gaussian and

$\mathbb{E}[\langle\xi^{U},\phi\rangle^{2}]=\int_{U\times
U}\gamma(x-y)\phi(x)\phi(y)\,\mathrm{d}x\,\mathrm{d}y.\qed$

######  Remark 5.10.

An example of a $\gamma$ that satisfies the conditions of Lemma 5.9 is the
following. Let $n\in\\{1,\dots,d\\}$ and $d_{1},\dots,d_{n}\in\mathbb{N}$ be
such that $d=d_{1}+\cdots+d_{n}$. Let
$\alpha_{1},\ldots,\alpha_{n}\in(0,\infty)$ are such that $\alpha_{j}<d_{j}$
for all $j$ and $\alpha_{1}+\ldots+\alpha_{n}<3$. Then, for
$x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{d_{1}}\times\cdots\times\mathbb{R}^{d_{n}}$,
we set

$\gamma(x)\vcentcolon=\lvert x_{1}\rvert^{-\alpha_{1}}\cdots\lvert
x_{n}\rvert^{-\alpha_{n}}.$

For this example, $f=\gamma$ and $g=0$. Observe that for the local
integrability of $f$, the condition $\alpha_{j}<d_{j}$ for all $j$ is
necessary.

######  Remark 5.11.

The 2D white noise $\xi^{2\textnormal{D}}$ does not satisfy the conditions of
Lemma 5.9. However, one has

$\mathbb{E}[\langle\xi_{\varepsilon_{1}}^{2\textnormal{D}}-\xi_{\varepsilon_{2}}^{2\textnormal{D}},\mathbbm{1}_{U}\phi^{\lambda}_{x}\rangle^{2}]=\lambda^{-2}\lVert(\mathbbm{1}_{U_{x}^{\lambda}}\phi)\ast(\rho_{\varepsilon_{1}/\lambda}-\rho_{\varepsilon_{2}/\lambda})\rVert_{L^{2}(\mathbb{R}^{d})},$

where $U_{x}^{\lambda}$ is as in (41). Therefore, the 2D white noise satisfies
Assumption 5.7 as well.

######  Lemma 5.12.

Assume 5.7(i). If the estimate (39) holds, then there exists a $\xi^{U}$ with
values in $\mathcal{C}^{-2+\delta}(\mathbb{R}^{d})$ such that the
distributions $\mathbbm{1}_{U}\xi_{\varepsilon}$ converge as follows, for all
$p\in(\frac{d}{\delta^{\prime}}+1,\infty)$,

$\displaystyle\lim_{\varepsilon\downarrow
0}\lVert\mathbbm{1}_{U}\xi_{\varepsilon}-\xi^{U}\rVert_{L^{p}(\mathbb{P},\mathcal{C}^{-2+\delta}(\mathbb{R}^{d}))}=0.$

If, furthermore, also the estimate (38) holds, then there exists
$r\in(0,\infty)$ such that for all $p\in(\frac{d}{\delta^{\prime}}+1,\infty)$,

$\mathbb{E}[\lVert\xi^{U}\rVert_{\mathcal{C}^{-2+\delta}(\mathbb{R}^{d})}^{p}]\lesssim_{p,\delta}\lvert
B(U,r)\rvert C^{\partial}_{p}.$ (42)

###### Proof.

Let $\delta_{+}\in(\delta,\delta+\delta^{\prime})$. We need the wavelet
characterization of Besov spaces given in Proposition C.31. Using the notation
therein, we have

$\mathbb{E}[\lVert\mathbbm{1}_{U}\xi_{\varepsilon_{1}}-\mathbbm{1}_{U}\xi_{\varepsilon_{2}}\rVert_{B_{p,p}^{-2+\delta_{+}}}^{p}]\\\
\lesssim_{p,\delta_{+}}\sum_{n\in\mathbb{N}_{0}}2^{np(-2+\delta_{+})}2^{-nd}\sum_{G\in\mathfrak{G}^{n},m\in\mathbb{Z}^{d}}\mathbb{E}[\lvert\langle\xi_{\varepsilon_{1}}-\xi_{\varepsilon_{2}},\mathbbm{1}_{U}2^{nd/2}\Psi^{n,G}_{m}\rangle\rvert^{p}]$

Since $\psi_{\mathfrak{f}}$ and $\psi_{\mathfrak{m}}$ are compactly supported,
there exists an $r\in(0,\infty)$ such that the sum with respect to $m$ is over
$\mathbb{Z}^{d}\cap 2^{n}B(U,r)$. Therefore, by (39), as $\sum_{m\in
2^{n}B(U,r)}\lesssim 2^{nd}|B(U,r)|$,

$\displaystyle\mathbb{E}[\lVert\mathbbm{1}_{U}\xi_{\varepsilon_{1}}-\mathbbm{1}_{U}\xi_{\varepsilon}\rVert_{B_{p,p}^{-2+\delta_{+}}}^{p}]$
$\displaystyle\lesssim_{p,\delta_{+}}\sum_{n\in\mathbb{N}_{0}}2^{-n(\delta+\delta^{\prime}-\delta_{+})}2^{-nd}\sum_{m\in
2^{n}B(U,r)}\bm{\varepsilon}_{p}^{\partial}(\varepsilon_{1},\varepsilon_{2};2^{-n},2^{-\max\\{n-1,0\\}m}).$
(43)

Because
$\sum_{n\in\mathbb{N}_{0}}2^{-n(\delta+\delta^{\prime}-\delta_{+})}2^{-nd}\sum_{m\in
2^{n}B(U,r)}1\lesssim|B(U,r)|\sum_{n\in\mathbb{N}_{0}}2^{-n(\delta+\delta^{\prime}-\delta_{+})}$,
in view of 5.7(i), the dominated convergence theorem yields that the right-
hand side (and thus left-hand side) of (43) converges to $0$. Now the
convergence of $\mathbbm{1}_{U}\xi_{\varepsilon}$ follows by the Besov
embedding (60) of Lemma A.1 (we may choose $\kappa>0$ small enough such that
$\frac{d}{p}+\kappa<\delta_{+}-\delta$). By using (38), one can similarly
prove the estimate (42). ∎

Recall the notation $G_{N}$ ($N\in\mathbb{N}_{0}$) from Definition 3.1.

######  Proposition 5.13.

Assume 5.7. Let $U$ be a bounded Lipschitz domain, $\sigma\in(0,\infty)$ and

$p\in(d/\sigma+2,\infty).$

Let $\tilde{Y}^{U}_{\varepsilon}$ be the distribution (see Definition 4.4 for
the notation)

$\varphi\mapsto\int_{\partial
U}\varphi\nabla(G_{0}\ast\xi_{\varepsilon})\cdot\,\mathrm{d}\bm{S}.$

Then there exists a $\tilde{Y}^{U}$ in $B^{-2+\delta}_{p,p}(\mathbb{R}^{d})$
such that
$\lVert\tilde{Y}^{U}_{\varepsilon}-\tilde{Y}^{U}\rVert_{B^{-2+\delta}_{p,p}(\mathbb{R}^{d})}\xrightarrow{\varepsilon\downarrow
0}0$ in probability. Furthermore, for every $m\in(0,\infty)$, uniformly over
$L\geq 1$ one has

$\mathbb{E}[\lVert\tilde{Y}^{U_{L}}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})}^{m}]\lesssim_{p,\delta,\sigma,U,m}L^{2m\sigma}\mathbb{E}[\lVert\xi\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}^{m}]+L^{d}C_{m}^{\partial}.$

###### Proof.

By the integration by parts formula (Lemma 4.3),

$\langle\tilde{Y}^{U_{L}}_{\varepsilon},\varphi\rangle=\int_{\partial
U_{L}}\varphi\nabla(G_{0}\ast\xi_{\varepsilon})\cdot\,\mathrm{d}\bm{S}=\int_{U_{L}}\nabla\varphi\cdot\nabla
G_{0}\ast\xi_{\varepsilon}-\int_{U_{L}}\varphi\Delta
G_{0}\ast\xi_{\varepsilon}.$ (44)

We first consider the first term. Let $\phi$ be a smooth function on
$\mathbb{R}^{d}$ such that $\phi=1$ on a neighborhood $U$. The map

$\mathcal{S}(\mathbb{R}^{d})\rightarrow\mathbb{R},\quad\varphi\mapsto\langle\mathbbm{1}_{U_{L}}\nabla\varphi,\phi(L^{-1}\cdot)\nabla(G_{0}\ast\xi)\rangle$
(45)

is well-defined, is independent of $\phi$ and is an element of
$B_{p,p}^{-2+\delta}(\mathbb{R}^{d})$. Indeed, if $q\in(1,\infty)$ is such
that $p^{-1}+q^{-1}=1$, by the duality [63, Theorem 2.17],

$\lvert\langle\mathbbm{1}_{U_{L}}\nabla\varphi,\phi(L^{-1}\cdot)\nabla(G_{0}\ast\xi)\rangle\rvert\leq\lVert\mathbbm{1}_{U_{L}}\nabla\varphi\rVert_{B_{q,q}^{1-\delta}(\mathbb{R}^{d})}\lVert\phi(L^{-1}\cdot)\nabla(G_{0}\ast\xi)\rVert_{B_{p,p}^{-1+\delta}(\mathbb{R}^{d})}.$

By Lemma A.6 and Lemma A.15 (see Definition A.11 for
$C_{\operatorname{Mult}}$), as
$1-\delta<\frac{1}{2}<1-\frac{1}{p}=\frac{1}{q}$,

$\lVert\mathbbm{1}_{U_{L}}\nabla\varphi\rVert_{B_{q,q}^{1-\delta}(\mathbb{R}^{d})}\lesssim_{q,\delta}C_{\operatorname{Mult}}^{U_{L}}[W_{q}^{1-\delta}]\lVert\nabla\varphi\rVert_{B_{q,q}^{1-\delta}(\mathbb{R}^{d})}\lesssim_{q,\delta,U}\lVert\varphi\rVert_{B_{q,q}^{2-\delta}(\mathbb{R}^{d})}.$

By Lemma A.1 (remember that we have $p\sigma>d$) and Lemma A.4,

$\lVert\phi(L^{-1}\cdot)\nabla(G_{0}\ast\xi)\rVert_{B_{p,p}^{-1+\delta}(\mathbb{R}^{d})}\lesssim_{p,\delta,\sigma}L^{2\sigma}\lVert\nabla(G_{0}\ast\xi)\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}.$

By Lemma A.6 and Corollary A.10,

$\lVert\nabla(G_{0}\ast\xi)\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}\lesssim_{\delta,\sigma}\lVert\xi\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}.$

Therefore, the distribution defined by (45) belongs to the dual space of
$B_{q,q}^{2-\delta}(\mathbb{R}^{d})$, which is identified with
$B_{p,p}^{-2+\delta}(\mathbb{R}^{d})$, and its norm in
$B_{p,p}^{-2+\delta}(\mathbb{R}^{d})$ is bounded by

$C_{p,\delta,\sigma,U}L^{2\sigma}\lVert\xi\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}.$

Now it is easy to see that this distribution is the limit of the first term of
the right-hand side of (44) (as
$\lVert\xi_{\varepsilon}-\xi\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}\rightarrow
0$).

Now we consider the second term of the right-hand side of (44). Note

$\int_{U}\varphi\Delta
G_{0}\ast\xi_{\varepsilon}=-\int_{U}\varphi\xi_{\varepsilon}+\int_{U}\varphi[\Delta(G_{0}-G)]\ast\xi_{\varepsilon}.$

The second term converges to

$\int_{U}\varphi[\Delta(G_{0}-G)]\ast\xi,$

as $[\Delta(G_{0}-G)]\ast\xi$ is a smooth function. The convergence and an
estimate of the first term is provided by Lemma 5.12. ∎

###  5.3 The Neumann Anderson Hamiltonian

As described in the beginning of Section 4 (below Lemma 4.5), the boundary
term will be dealt with by the decomposition into symmetric forms
$\tilde{\mathcal{Z}}$ and $\widehat{\mathcal{Z}}$. Let us first consider the
ingredients for the latter symmetric form.

######  Definition 5.14.

Let $U$ be a bounded Lipschitz domain. For $N\in\mathbb{N}_{0}$ we define

$\displaystyle\widehat{Y}_{N}$
$\displaystyle\vcentcolon=G_{N}\ast(X-\xi)+(G_{N}-G_{0})\ast\xi,$
$\displaystyle\widehat{Y}^{\varepsilon}_{N}$
$\displaystyle\vcentcolon=G_{N}\ast(X^{\varepsilon}-\xi_{\varepsilon})+(G_{N}-G_{0})\ast\xi_{\varepsilon},\qquad\varepsilon\in(0,1).$

######  Lemma 5.15.

Let $U$ be a bounded Lipschitz domain. Then for $\delta\in(\frac{1}{2},1)$ and
$N\in\mathbb{N}_{0}$

$\displaystyle\lVert\widehat{Y}_{N}\rVert_{\mathcal{C}^{1+\delta,\sigma}}$
$\displaystyle\lesssim_{U,\delta}\lVert
X-\xi\rVert_{\mathcal{C}^{-2+2\delta,\sigma}(\mathbb{R}^{d})}+2^{N}\lVert\xi\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}.$

In particular,
$\lVert\widehat{Y}_{N}-\widehat{Y}^{\varepsilon}_{N}\rVert_{\mathcal{C}^{1+\delta,\sigma}}$
converges in probability to $0$.

###### Proof.

The estimate and a similar one for
$\lVert\widehat{Y}_{N}-\widehat{Y}^{\varepsilon}_{N}\rVert_{\mathcal{C}^{1+\delta,\sigma}}$
from which the convergence follows by Theorem 3.3, follow from Corollary A.10.
∎

######  Definition 5.16.

Assume 3.10 and 5.7, let $U$ be a bounded Lipschitz domain. Let
$r\in(1-\delta,1)$. Using the notation of Theorem 4.6 and of Proposition 5.13,
we define the following symmetric forms on $H^{r}(U)$, for
$N\in\mathbb{N}_{0}$

$\displaystyle\tilde{\mathcal{Z}}_{N}[U]$
$\displaystyle\vcentcolon=\tilde{\mathcal{Z}}_{\tilde{Y}^{U},e^{2W_{N}}}^{U},\qquad\tilde{\mathcal{Z}}^{\varepsilon}_{N}[U]\vcentcolon=\tilde{\mathcal{Z}}_{\tilde{Y}^{U}_{\varepsilon},e^{2W^{\varepsilon}_{N}}}^{U},$
$\displaystyle\widehat{\mathcal{Z}}_{N}[U]$
$\displaystyle\vcentcolon=\widehat{\mathcal{Z}}^{U}_{\widehat{Y}_{N},e^{2W_{N}}},\qquad\widehat{\mathcal{Z}}^{\varepsilon}_{N}[U]\vcentcolon=\widehat{\mathcal{Z}}^{U}_{\widehat{Y}^{\varepsilon}_{N},e^{2W^{\varepsilon}_{N}}}.$

We furthermore make abuse of notation (compared to the symmetric forms on
$H_{0}^{r}(U)$ as in (30)) and define the following symmetric forms on
$H^{r}(U)$,

$\mathcal{Z}_{N}[U]\vcentcolon=\mathcal{Z}^{U}_{Y_{N}},\quad\mathcal{Z}^{\varepsilon}_{N}[U]\vcentcolon=\mathcal{Z}^{U}_{Y^{\varepsilon}_{N}}.$
(46)

Then we define

$\displaystyle\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}}}_{N}[U]$
$\displaystyle\vcentcolon=\mathcal{Z}_{N}[U]+\tilde{\mathcal{Z}}_{N}[U]+\widehat{\mathcal{Z}}_{N}[U],$
$\displaystyle\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}},\varepsilon}_{N}[U]$
$\displaystyle\vcentcolon=\mathcal{Z}^{\varepsilon}_{N}[U]+\tilde{\mathcal{Z}}^{\varepsilon}_{N}[U]+\widehat{\mathcal{Z}}^{\varepsilon}_{N}[U],\qquad\varepsilon\in(0,1).$

Recalling the notations from Definition 4.9 and Proposition 4.13, for
$M=M(U,\delta;1)$ (see (31)) we set

$\displaystyle\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}}}$
$\displaystyle\vcentcolon=\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}\vcentcolon=\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}_{W_{M},\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}}}_{M}[U]},$
$\displaystyle\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U)$
$\displaystyle\vcentcolon=\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U;W_{M},\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}}}_{M}[U]),$
$\displaystyle\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}$
$\displaystyle\vcentcolon=\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}},U}\vcentcolon=\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}}}_{W^{\varepsilon}_{M},\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}},\varepsilon}_{M}[U]},$
$\displaystyle\lambda_{k;\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}(U)$
$\displaystyle\vcentcolon=\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U;W^{\varepsilon}_{M},\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}},\varepsilon}_{M}[U]).$

######  Theorem 5.17.

Assume 3.10 and 5.7. Let $U$ be a bounded Lipschitz domain. For
$\varepsilon\in(0,1)$,

$\displaystyle\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}=-\Delta-\xi_{\varepsilon}+c_{\varepsilon}$
(47)

Then, one has (see Definition 1.5 for “$\xrightarrow{\textnormal{NR}}$”)

$\displaystyle\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}},U}$
$\displaystyle\xrightarrow{\textnormal{NR}}_{\varepsilon\downarrow
0}\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}\mbox{ in probability,}$
$\displaystyle\lambda_{k;\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}(U)$
$\displaystyle\xrightarrow{\varepsilon\downarrow
0}\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U)\mbox{ in probability,}\qquad
k\in\mathbb{N}.$

and for all $k\in\mathbb{N}$ there exist an eigenfunction $\phi_{k}$ of
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}$ corresponding to
$\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U)$ and eigenfunctions $\phi_{k,m}$
of $\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}_{\varepsilon}$ corresponding to
$\lambda_{k;\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}(U)$ for all
$\varepsilon>0$ such that

$\displaystyle\phi_{k,m}\xrightarrow{m\to\infty}\phi_{k}\quad\mbox{ in
}L^{2}(U)\mbox{ in probability}.$

###### Proof.

(47) follows by Theorem 3.3 and Lemma 4.5.

The rest of the proof is similar to that of Theorem 5.4 and uses Proposition
5.13 and Lemma 5.15. ∎

######  Lemma 5.18.

Assume 3.10. Let $U$ be a bounded Lipschitz domain and

$\frac{1}{2}<\delta_{-}<\delta.$

Let $r\in(1-\delta_{-},1)$. For $\gamma\in(0,\infty)$, we define
$M_{L,\delta_{-},\gamma}\vcentcolon=M(U_{L},\delta_{-};\gamma)$.

  1. (a)

Then, for every $\sigma\in(0,\infty)$, $L\geq 1$ and $\gamma\in(0,\infty)$,

$\llbracket\mathcal{Z}_{M_{L,\delta_{-},\gamma}}[U_{L}]\rrbracket_{H^{r}(U_{L})}\lesssim_{U,\delta_{-},\delta,\sigma}\gamma^{-(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}L^{(2+\bm{\mathfrak{b}}(\delta-\delta_{-})^{-1})\sigma}\lVert
X\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}^{(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}\bm{\mathfrak{a}}.$

  2. (b)

Additionally, assume 5.7. Suppose that $r\in(\frac{3}{2}-\delta_{-},1)$,
$\varepsilon\in(0,\delta_{-}-\frac{1}{2})$, $p\in(2,\infty)$ and $q\in(1,2)$
are such that $\frac{1}{p}+\frac{1}{q}=1$ and (17) holds for $s=r$. Then, for
every $\sigma\in(0,\infty)$, $L\geq 1$ and $\gamma\in(0,1]$,

$\llbracket\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}}}_{M_{L,\delta_{-},\gamma}}[U_{L}]\rrbracket_{H^{r}(U_{L})}\\\
\lesssim_{U,\delta_{-},\delta,\sigma,\varepsilon}\gamma^{-(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}L^{(2+(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}})\sigma}\lVert
X\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}^{(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}(\bm{\mathfrak{a}}+\lVert\xi\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})})\\\
+L^{2\varepsilon}\lVert\tilde{Y}^{U_{L}}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})}+L^{\sigma}\lVert
X-\xi\rVert_{\mathcal{C}^{-2+2\delta,\sigma}(\mathbb{R}^{d})}.$

###### Proof.

(a) follows as in the proof of Lemma 5.6 by (36) and (34).

For (b) we use (a) and estimate $\tilde{\mathcal{Z}}_{M_{L,\gamma}}[U_{L}]$
and $\widehat{\mathcal{Z}}_{M_{L,\gamma}}[U_{L}]$. By Theorem 4.6 (b) and (c)
we have

$\displaystyle\tilde{\mathcal{Z}}_{M_{L,\gamma}}[U_{L}]$
$\displaystyle\lesssim_{\delta,\varepsilon,p,U}L^{2\varepsilon}\lVert
e^{2W_{M_{L,\gamma}}}\rVert_{C^{\delta}(U_{L})}\lVert\tilde{Y}^{U_{L}}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})},$
$\displaystyle\widehat{\mathcal{Z}}_{M_{L,\gamma}}[U_{L}]$
$\displaystyle\lesssim_{\delta,\sigma,U}L^{\sigma}\lVert
e^{2W_{M_{L,\gamma}}}\rVert_{C^{\delta}(U_{L})}\lVert\widehat{Y}_{M_{L,\gamma}}\rVert_{\mathcal{C}^{1+{\delta},\sigma}(\mathbb{R}^{d})}.$

As for any $x,y\in\mathbb{R}$,

$\displaystyle|e^{x}-e^{y}|=e^{x}|1-e^{y-x}|\leq Ce^{x}|y-x|\leq Ce^{x\vee
y}|y-x|,$

by definition of $M_{L,\gamma}$ we have $\lVert
e^{2W_{M_{L,\gamma}}}\rVert_{\mathcal{C}^{\delta}(U)}\lesssim 2\gamma
e^{2\gamma}\leq 2e^{2}$.

Therefore, we obtain the desired inequality by the estimate of
$\lVert\widehat{Y}_{M_{L,\gamma}}\rVert_{\mathcal{C}^{1+\delta,\sigma}(\mathbb{R}^{d})}$
from Lemma 5.15 and (34). ∎

Without Assumption 5.7, we can still construct an artificial Neumann Anderson
Hamiltonian, which will be used in Section 5.4 as a technical tool.

######  Definition 5.19.

Assume 3.10 and let $U$ be a bounded Lipschitz domain and $r\in(1-\delta,1)$.
For $M=M(U,\delta;1)$, we set

$\overline{\mathcal{H}}^{\scaleobj{0.8}{\mathrm{N}},U}\vcentcolon=\mathcal{H}_{W_{M},\mathcal{Z}_{M}[U]}^{\scaleobj{0.8}{\mathrm{N}},U},\quad\overline{\lambda}^{\scaleobj{0.8}{\mathrm{N}}}_{k}(U)\vcentcolon=\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U;W_{M},\mathcal{Z}_{M}[U]).$

######  Remark 5.20.

Similar to Remark 5.5, for $\mathfrak{M}$ being a random variable with values
in $\mathbb{N}_{0}$ such that $\mathfrak{M}\geq M(U,\delta,1)$, one has,
almost surely,
$\overline{\mathcal{H}}^{\scaleobj{0.8}{\mathrm{N}},U}=\overline{\mathcal{H}}^{\scaleobj{0.8}{\mathrm{N}},U}_{W_{\mathfrak{M}},\mathcal{Z}_{\mathfrak{M}}[U]}$
and, under Assumption 5.7,
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}={\mathcal{H}}^{\scaleobj{0.8}{\mathrm{N}},U}_{W_{\mathfrak{M}},\mathcal{Z}_{\mathfrak{M}}[U]}$.

######  Remark 5.21.

To construct the natural Neumann Anderson Hamiltonian without Assumption 5.7,
we conjecture that “boundary renormalisation” is necessary. For instance, if
$\xi$ is the 3D white noise, the recent work [30] suggests that the operators
associated to the symmetric forms

$(u,v)\mapsto\int_{U}\nabla u\cdot\nabla
v-(\xi_{\varepsilon}-c_{\varepsilon})uv+c^{\prime}_{\varepsilon}\int_{\partial
U}uv\,\mathrm{d}S$

converge, where the constants $c^{\prime}_{\varepsilon}$ diverge
logarithmically.

###  5.4 Integrated density of states

The aim of this section is to construct the integrated density of states
associated to the Anderson Hamiltonian with potential $\xi$. For this sake, we
need the following assumption.

######  Assumption 5.22 (Assumption III).

Recall that our probability space $\Omega$ is the space
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ of tempered distributions. Then, we
have maps $T_{x}:\Omega\rightarrow\Omega$ ($x\in\mathbb{R}^{d}$) of
translations $\omega\to\omega(\cdot-x)$. We assume the probability measure
$\mathbb{P}$ to be ergodic with respect to $(T_{x})_{x\in\mathbb{R}^{d}}$.

######  Lemma 5.23 ([61, Proposition 6.1]).

Let $\xi$ be a centered Gaussian field such that

$\lim_{x\to\infty}\mathbb{E}[\xi(0)\xi(x)]=0.$

Then, Assumption 5.22 holds.

######  Assumption 5.24.

(Assumptions for this section) Throughout this section, we assume 3.10 and
5.22. We fix $\delta_{-},\delta,r\in(0,1)$ and $p\in(2,\infty)$ such that
$\delta_{-}<\delta$ and (17) is satisfied for $q\in(1,2)$ such that
$\frac{1}{p}+\frac{1}{q}=1$ and some
$\varepsilon\in(0,\delta_{-}-\frac{1}{2})$. We also fix
$\sigma\in(0,\frac{1}{4})$ satisfying

$(2+(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}(\delta_{-}))d\sigma<1-r.$ (48)

We set

$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert\vcentcolon=1+\lVert
X\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}+\bm{\mathfrak{a}}+\lVert
X-\xi\rVert_{\mathcal{C}^{-2+2\delta,\sigma}(\mathbb{R}^{d})},$
$\displaystyle\bm{\mathfrak{a}}^{\varepsilon}\vcentcolon=\bm{\mathfrak{a}}^{\varepsilon}(\delta,\sigma)\vcentcolon=\sup_{N\in\mathbb{N}}2^{-\bm{\mathfrak{b}}N}\lVert
Y^{\varepsilon}_{N}\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}\in
L^{p}(\mathbb{P}),$
$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi_{\varepsilon}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert\vcentcolon=1+\lVert
X^{\varepsilon}\rVert_{\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})}+\bm{\mathfrak{a}}^{\varepsilon}+\lVert
X^{\varepsilon}-\xi_{\varepsilon}\rVert_{\mathcal{C}^{-2+2\delta,\sigma}(\mathbb{R}^{d})},\qquad\varepsilon>0.$

Whenever we assume 5.7, we implicitly also assume $\delta_{-}>\frac{1}{2}$;
and for any bounded Lipshitz domain $U$, we set (with $\tilde{Y}^{U}$ as in
Proposition 5.13)

$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\xi\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\partial
U}\vcentcolon=\sup_{L\in\mathbb{N}}L^{-\frac{1}{4}}\lVert\tilde{Y}^{U_{L}}\rVert_{B_{p,p}^{-1+\delta}(\mathbb{R}^{d})},\quad\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\xi_{\varepsilon}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\partial
U}\vcentcolon=\sup_{L\in\mathbb{N}}L^{-\frac{1}{4}}\lVert\tilde{Y}^{U_{L}}_{\varepsilon}\rVert_{B_{p,p}^{-1+\delta}(\mathbb{R}^{d})},\quad\varepsilon>0.$

######  Remark 5.25.

By Theorem 3.3,
$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert\in
L^{q}(\mathbb{P})$ for every $q\in[1,\infty)$ and under Assumption 5.7, by
Proposition 5.13,
$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\xi\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\partial
U}\in L^{q}(\mathbb{P})$ for every $q\in[1,\infty)$. By Lemma 5.6, Lemma 5.18
and the condition on $\sigma$, (48), there exists an $m\in\mathbb{N}$ such
that for all bounded Lipschitz domains $U$, for all $L\geq 1$ and
$\gamma\in(0,1]$, for $M_{L,\gamma}=M(U_{L},\delta_{-},\gamma)$,

$\displaystyle\llbracket\mathcal{Z}_{M_{L,\gamma}}[U_{L}]\rrbracket_{H^{r}(U_{L})}$
$\displaystyle\lesssim_{U}\gamma^{-(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}L^{\frac{1-r}{d}}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{m},$
(49)
$\displaystyle\llbracket\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}}}_{M_{L,\gamma}}[U_{L}]\rrbracket_{H^{r}(U_{L})}$
$\displaystyle\lesssim_{U}\gamma^{-(\delta-\delta_{-})^{-1}\bm{\mathfrak{b}}}L^{\frac{1-r}{d}}(\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{m}+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\xi\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\partial
U})\quad\mbox{ under Assumption\leavevmode\nobreak\ \ref{assump:boundary}}.$
(50)

(For (50) observe that $\sigma\leq\frac{1-r}{d}$ and that we may choose
$\varepsilon>0$ as in Lemma 5.18 (b) such that $2\varepsilon<\frac{1-r}{d}$.)
In (49) one may replace “$\mathcal{Z}$” and “$\bm{\xi}$” by
“$\mathcal{Z}^{\varepsilon}$” and “$\bm{\xi_{\varepsilon}}$” and in (50) one
may replace “$\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}}}$”, “$\bm{\xi}$” and
“$\xi$” by “$\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}},\varepsilon}$”,
“$\bm{\xi_{\varepsilon}}$” and “$\xi_{\varepsilon}$”.

######  Definition 5.26.

Recall the notation from Definition 4.20. For a bounded domain $U$,
$\lambda\in\mathbb{R}$ and $\varepsilon>0$, we set
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}\vcentcolon=\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}_{W_{M},\mathcal{Z}_{M}}$
and
$\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}\vcentcolon=\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}_{W^{\varepsilon}_{M},\mathcal{Z}^{\varepsilon}_{M}}$,
i.e.,

$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)\vcentcolon=\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U)\leq\lambda\\}},\qquad\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)\vcentcolon=\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\lambda_{k;\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}(U)\leq\lambda\\}}.$

If $U$ is a bounded Lipshitz domain, we set
$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}\vcentcolon=\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{W_{M},\mathcal{Z}_{M}}$
and
$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}_{\varepsilon}\vcentcolon=\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{W^{\varepsilon}_{M},\mathcal{Z}^{\varepsilon}_{M}}$,
i.e.,

$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\vcentcolon=\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\overline{\lambda}^{\scaleobj{0.8}{\mathrm{N}}}_{k}(U)\leq\lambda\\}},\qquad\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\vcentcolon=\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\overline{\lambda}^{\scaleobj{0.8}{\mathrm{N}}}_{k;\varepsilon}(U)\leq\lambda\\}},$

and under Assumption 5.7 we set
$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}\vcentcolon=\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{W_{M},\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}}}_{M}}$
and
$\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}\vcentcolon=\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{W^{\varepsilon}_{M},\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}},\varepsilon}_{M}}$,
i.e.,

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\vcentcolon=\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U)\leq\lambda\\}},\qquad\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\vcentcolon=\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\lambda_{k;\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}(U)\leq\lambda\\}}.$

######  Remark 5.27.

In most of the following we restrict our statements to
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}$,
$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$ and
$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}$. However, by ‘adding some
$\varepsilon$’s’ the statements are also valid by replacing the occurrences of
“$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}$”,
“$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$”
“$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}$”, “$\bm{\xi}$” and “$\xi$”, and by
“$\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}$”,
“$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}_{\varepsilon}$”
“$\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}$”,
“$\bm{\xi}_{\varepsilon}$” and “$\xi_{\varepsilon}$”.

######  Lemma 5.28.

Let $U$ be a bounded Lipschitz domain. Then, for any $\theta\in(0,1)$, there
exist $\lambda_{U,\theta}$, $C_{U,\theta,r}\in(0,\infty)$ and an integer
$l\in\mathbb{N}$ such that for every $\lambda\geq\lambda_{\theta,U}$

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)$
$\displaystyle\geq(1-\theta)\frac{\lvert B(0,1)\rvert}{(2\pi)^{d}}\lvert
U\rvert\\{\lambda+\theta+C_{U,\theta,r}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{l}\\}^{\frac{d}{2}},$
(51) $\displaystyle\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)$
$\displaystyle\leq(1+\theta)\frac{\lvert B(0,1)\rvert}{(2\pi)^{d}}\lvert
U\rvert\\{\lambda+\theta+C_{U,\theta,r}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{l}\\}^{\frac{d}{2}}.$
(52)

In particular,
$\mathbb{E}[\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)^{m}]<\infty$
for every $m\in(0,\infty)$ and $\lambda\in\mathbb{R}$.

If we furthermore assume 5.7, then

$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\leq(1+\theta)\frac{\lvert
B(0,1)\rvert}{(2\pi)^{d}}\lvert
U\rvert\\{\lambda+\theta+C_{U,\theta,r}(\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\xi\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\partial
U})^{l}\\}^{\frac{d}{2}}.$ (53)

###### Proof.

The proof of (51) and (53) are similar to (52), hence we only give the proof
of the latter. Remember that $\bm{N}_{0}^{\scaleobj{0.8}{\mathrm{D}}}$ and
$\bm{N}_{0}^{\scaleobj{0.8}{\mathrm{N}}}$ are the eigenvalue counting
functions of $-\Delta$ with Dirichlet and Neumann boundary conditions,
respectively. By Lemma 4.23 (b), there exists a $\lambda_{U,\theta}>0$ such
that for $\lambda\geq\lambda_{U,\theta}$ we have

$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{0}(U,\lambda)\leq(1+\theta)^{\frac{1}{2}}\frac{\lvert
B(0,1)\rvert}{(2\pi)^{d}}\lvert U\rvert\lambda^{\frac{d}{2}}.$

Let $\theta^{\prime}\in(0,\infty),\gamma\in(0,1]$. By Lemma 4.22, and Remark
5.20, with
$\Lambda^{+}_{\lambda,\theta^{\prime}}\vcentcolon=\Lambda^{+}_{\lambda,\theta^{\prime}}(W_{M_{\gamma}},\mathcal{Z}_{M_{\gamma}})$
where $M_{\gamma}$ is the random variable $M(U,\delta;\gamma)$ (see (31)), one
has

$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\leq(1+\theta)^{\frac{1}{2}}\frac{\lvert
B(0,1)\rvert}{(2\pi)^{d}}\lvert
U\rvert(\Lambda^{+}_{\lambda,\theta^{\prime}})^{\frac{d}{2}}.$

Recalling the definition of $\Lambda^{+}_{\lambda,\theta^{\prime}}$, one
observes that there exists a constant $C^{\prime}_{U,\theta^{\prime},r}$ such
that

$\Lambda^{+}_{\lambda,\theta^{\prime}}\leq(1+\theta^{\prime})e^{(2+\frac{2r}{1-r})\gamma}(\lambda+\theta^{\prime}+C^{\prime}_{U,\theta^{\prime},r}\llbracket\mathcal{Z}_{M_{\gamma}}\rrbracket_{H^{r}(U)}^{\frac{1}{1-r}}).$

Therefore, if
$\gamma\vcentcolon=(2+\frac{2r}{1-r})^{-1}\log(1+\theta^{\prime})$ and
$\theta^{\prime}\vcentcolon=(1+\theta)^{\frac{1}{2d}}-1\in(0,\theta)$, and
$C_{U,\theta,r}=C^{\prime}_{U,\theta^{\prime},r}\gamma^{\frac{1-r}{d\sigma}-2}$
(see (48)), using (49), one has

$\displaystyle\Lambda^{+}_{\lambda,\theta^{\prime}}$
$\displaystyle\leq(1+\theta)^{\frac{1}{d}}(\lambda+\theta+C_{U,\theta,r}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{l}),$

which yields (52). ∎

By Lemma 4.22 we derive
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)\leq\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)$
and similarly - by using that
$\mathcal{Z}_{N}^{\scaleobj{0.8}{\mathrm{N}}}[U]=\mathcal{Z}_{N}[U]$ on
$H_{0}^{r}$ (which follows due to the fact that $\mathcal{T}$ equals zero on
$H_{0}^{r}(U)$, see Lemma 2.7) -
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)$.
Therefore, as a direct consequence of Lemma 5.28 we obtain the following
asymptotics. These asymptotics agree with the asymptotics of the eigenvalue
counting function for the Laplacian operator, as proven by Weyl (also called
Weyl’s law) and later generalised for a class of Schrödinger operators by
Kirsch and Martinelli [42, Proposition 2.3] (observe that our results agree
with the work of Mouzard on two dimensional manifolds, see [57]).

######  Proposition 5.29.

Let $U$ be a bounded Lipschitz domain, then

$\displaystyle\lim_{\lambda\rightarrow\infty}\lambda^{-\frac{d}{2}}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)=\lim_{\lambda\rightarrow\infty}\lambda^{-\frac{d}{2}}\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)=\frac{|B(0,1)|}{(2\pi)^{d}}|U|,$

and under Assumption 5.7,

$\displaystyle\lim_{\lambda\rightarrow\infty}\lambda^{-\frac{d}{2}}\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)=\frac{|B(0,1)|}{(2\pi)^{d}}|U|.$

######  Proposition 5.30.

There exist (deterministic) functions $\mathbb{R}\rightarrow[0,\infty)$,
$\bm{\mathcal{N}}^{\scaleobj{0.8}{\mathrm{D}}}$ and
$\overline{\bm{\mathcal{N}}}^{\scaleobj{0.8}{\mathrm{N}}}$, such that for all
$\lambda\in\mathbb{R}$ and $y\in\mathbb{R}^{d}$, $\mathbb{P}$-almost surely
and in $L^{1}(\mathbb{P})$, with $Q=y+[-\frac{1}{2},\frac{1}{2}]^{d}$,

$\displaystyle\bm{\mathcal{N}}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)$
$\displaystyle=\lim_{L\to\infty}\frac{1}{L^{d}}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(Q_{L},\lambda),$
$\displaystyle\overline{\bm{\mathcal{N}}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda)$
$\displaystyle=\lim_{L\in\mathbb{Q},L\to\infty}\frac{1}{L^{d}}\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(Q_{L},\lambda)$

exist. Moreover,

$\displaystyle\sup_{L>0}\frac{1}{L^{d}}\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(Q_{L},\lambda)]$
$\displaystyle=\bm{\mathcal{N}}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)$
$\displaystyle\leq\overline{\bm{\mathcal{N}}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda)=\inf_{L>0}\frac{1}{L^{d}}\mathbb{E}[\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(Q_{L},\lambda)].$
(54)

Under Assumption 5.7, one may simultaneously replace
$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$ by
$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}$ and
$\overline{\bm{\mathcal{N}}}^{\scaleobj{0.8}{\mathrm{N}}}$ by
$\bm{\mathcal{N}}^{\scaleobj{0.8}{\mathrm{N}}}$ in the above definition and
inequality.

###### Proof.

This follows by the ergodic theorem by Akcoglu and Krengel [1], see [42,
Section 3] for more details, for which applicability we use Assumption 5.22
and check the following. By Lemma 4.27 and Remark 4.28
$Q\mapsto\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(Q,\lambda)$ is superadditive and
$Q\mapsto\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(Q,\lambda)$ is
subadditive. Furthermore, if $Q\subseteq[-1,1]^{d}$, Lemma 4.21 implies

$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(Q,\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}([-1,1]^{d},\lambda)\leq\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}([-1,1]^{d},\lambda),$

and $\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}([-1,1]^{d},\lambda)$ is in
$L^{1}(\mathbb{P})$ by Lemma 5.28. ∎

######  Remark 5.31.

The cube $Q$ in Proposition 5.30 does not need to be centered at the origin.
This is important in the proof of Theorem 5.38.

######  Definition 5.32.

We define (deterministic) functions
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}},\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}:\mathbb{R}\rightarrow\mathbb{R}$
by

$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)\vcentcolon=\inf_{\lambda^{\prime}>\lambda}\bm{\mathcal{N}}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda^{\prime})\mbox{
and
}\quad\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda)\vcentcolon=\inf_{\lambda^{\prime}>\lambda}\overline{\bm{\mathcal{N}}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda^{\prime}).$

Note that they are right-continuous functions that satisfy
$\lim_{\lambda\rightarrow-\infty}\bm{N}^{\\#}(\lambda)=0$ for $\\#$ denoting
either $\scaleobj{0.8}{\mathrm{D}}$ or $\scaleobj{0.8}{\mathrm{N}}$.

######  Definition 5.33.

A sequence $(f_{n})_{n\in\mathbb{N}}$ of increasing functions
$\mathbb{R}\rightarrow[0,\infty)$, is said to _converge vaguely_ to some
function $f:\mathbb{R}\rightarrow[0,\infty)$ if $f_{n}(\lambda)\rightarrow
f(\lambda)$ for all $\lambda\in\mathbb{R}$ that are continuity points of $f$.

######  Remark 5.34.

If $\lambda$ is a continuity point of
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)$, then
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)=\bm{\mathcal{N}}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)$.

######  Remark 5.35.

Observe that by Lemma 4.31 (b) and (49) for $\gamma=1$, for all $\mu>0$ and
bounded Lipschitz domains $U$, there exists a $C_{U,\mu}>0$ such that

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda)$
$\displaystyle\leq\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U_{L},\lambda)$
$\displaystyle\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda+\mu)+C_{U,\mu}L^{d-\frac{1}{2}}[1+\max\\{\lambda,0\\}+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{\frac{m}{1-r}}]^{\frac{d}{2}},$

and under Assumption 5.7,

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda)$
$\displaystyle\leq\bm{N}(U_{L},\lambda)$
$\displaystyle\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda+\mu)+C_{U,\mu}L^{d-\frac{1}{2}}[1+\max\\{\lambda,0\\}+(\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{m}+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\xi\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\partial
U})^{\frac{1}{1-r}}]^{\frac{d}{2}}.$

######  Proposition 5.36.

$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}=\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$
and, under Assumption 5.7,
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}=\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}$.

###### Proof.

Let $\lambda$ be a continuity point of both
$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}$ and
$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$. By Remark 5.35 applied to
$U=[-\frac{1}{2},\frac{1}{2}]^{d}$, by Proposition 5.30, because
$\bm{\mathcal{N}}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)=\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)$
and
$\overline{\bm{\mathcal{N}}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda)=\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda)$,
see Remark 5.34, we have for all $\mu>0$,

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)\leq\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda+\mu),$

so that the equality follows as both $\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}$ and
$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$ are right-continuous. Under
Assumption 5.7 we can argue similarly with
“$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}$” instead of
“$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$”. ∎

######  Definition 5.37.

Thanks to Proposition 5.36, we may simply write

$\bm{N}\vcentcolon=\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}=\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(=\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}\quad\mbox{under
Assumption\leavevmode\nobreak\ \ref{assump:boundary}}).$

We call $\bm{N}$ the _integrated density of states_ for the Anderson
Hamiltonian with potential $\xi$.

######  Theorem 5.38.

Let $U$ be a bounded domain. Then, almost surely,

$\lim_{L\to\infty}\frac{1}{\lvert
U_{L}\rvert}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\cdot)=\bm{N}\quad\mbox{vaguely}.$
(55)

If $U$ is a bounded Lipschitz domain, then

$\lim_{L\in\mathbb{N},L\to\infty}\frac{1}{\lvert
U_{L}\rvert}\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(U_{L},\cdot)=\bm{N}\quad\mbox{vaguely}.$
(56)

Under Assumption 5.7, one can replace
$\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$ by
$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}$ in (56).

###### Proof.

Firstly, observe that we may assume $U$ to be a bounded Lipschitz domain due
to the monotonicity of $\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)$ as a
function of $U$. By Remark 5.35 it suffices to prove (55) (also for
$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}$ under Assumption 5.7).

Let $\lambda\in\mathbb{R}$ be a continuity point of $\bm{N}$. We set

$I_{n}\vcentcolon=\\{k\in\mathbb{Z}^{d}\nonscript\>|\nonscript\>\mathopen{}k+[0,1]^{d}\subseteq
2^{n}U\\},\quad
J_{n}\vcentcolon=\\{k\in\mathbb{Z}^{d}\nonscript\>|\nonscript\>\mathopen{}(k+[0,1]^{d})\cap(2^{n}U)\neq\varnothing\\}.$

By Lemma 4.21 and Lemma 4.27,

$\sum_{k\in
I_{n}}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(2^{-n}L(k+[0,1]^{d}),\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda)\leq\sum_{k\in
J_{n}}\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(2^{-n}L(k+[0,1]^{d}),\lambda).$

Therefore, by Proposition 5.30 and Remark 5.34,

$\frac{\\#I_{n}}{2^{dn}\lvert
U\rvert}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda)\leq\liminf_{L\to\infty}\frac{1}{\lvert
U_{L}\rvert}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda)\leq\limsup_{L\to\infty}\frac{1}{\lvert
U_{L}\rvert}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda)\leq\frac{\\#J_{n}}{2^{dn}\lvert
U\rvert}\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(\lambda).$

By Proposition 5.36, one has
$\bm{N}=\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}=\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}$.
Thus, the proof is complete by letting $n\to\infty$. ∎

######  Remark 5.39.

Recall Remark 5.27. Let $\varepsilon\in(0,1)$. There exist functions
$\bm{\mathcal{N}}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}},\overline{\bm{\mathcal{N}}}^{\scaleobj{0.8}{\mathrm{N}}}_{\varepsilon}$
and $\bm{\mathcal{N}}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}$ such that
analogues statements as in Proposition 5.30 hold. Then we define
$\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda):=\inf_{\lambda^{\prime}>\lambda}\bm{\mathcal{N}}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}(\lambda^{\prime})$
and similarly $\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}_{\varepsilon}$
and $\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}$. By analogous
arguments as in Theorem 5.38 we also have
$\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}=\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}_{\varepsilon}$
($=\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}}}$ under Assumption 5.7).
In this case
$\bm{N}_{\varepsilon}:=\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}$ is
called the integrated density of states for the Anderson Hamiltonian with
potential $\xi_{\varepsilon}-c_{\varepsilon}$.

For the convergence of $\bm{N}_{\varepsilon}$ to $\bm{N}$ as in Theorem 5.41,
we introduce the following auxiliary lemma.

######  Lemma 5.40.

Let $\\#$ denote either $\scaleobj{0.8}{\mathrm{D}}$ or
$\scaleobj{0.8}{\mathrm{N}}$. For all $L>0$, $\lambda\in\mathbb{R}$ and
$\mu>0$,

$\displaystyle\liminf_{\varepsilon\downarrow
0}\mathbb{E}[\bm{N}_{\varepsilon}^{\\#}(Q_{L},\lambda)]\geq\mathbb{E}[\bm{N}^{\\#}(Q_{L},\lambda-\mu)].$

###### Proof.

First we observe that as
$\lambda_{k;\varepsilon}^{\\#}(Q_{L})\to\lambda_{k}^{\\#}(Q_{L})$ in
probability for all $k$ (by Theorem 5.4 and Theorem 5.17), the following
holds: For all $(\varepsilon_{n})_{n\in\mathbb{N}}$ in $(0,\infty)$ with
$\varepsilon_{n}\downarrow 0$ as $n\rightarrow\infty$, there exists a
subsequence $(\varepsilon_{n_{m}})_{m\in\mathbb{N}}$ and a
$\Omega_{1}\subset\Omega$ of $\mathbb{P}$-probability $1$, such that on
$\Omega_{1}$,
$\lambda_{k;\varepsilon_{n_{m}}}^{\\#}(Q_{L})\to\lambda_{k}^{\\#}(Q_{L})$ for
all $k$, and therefore for all $\mu>0$

$\displaystyle\liminf_{\varepsilon\downarrow
0}\bm{N}_{\varepsilon}^{\\#}(Q_{L},\lambda)\geq\bm{N}^{\\#}(Q_{L},\lambda-\mu)$

(indeed, if $\mu_{1,m}\leq\mu_{2,m}\leq\cdots$ and
$\mu_{k,m}\rightarrow\mu_{k}$ in $\mathbb{R}$ as $m\rightarrow\infty$, for all
$k\in\mathbb{N}$, then
$\liminf_{m\rightarrow\infty}\sum_{k=1}^{\infty}\mathbbm{1}_{\\{\mu_{k,m}\leq\mu\\}}\geq\sum_{k=1}^{\infty}\mathbbm{1}_{\\{\mu_{k}\leq\mu-\varepsilon\\}}$
for all $\mu\in\mathbb{R}$ and $\varepsilon>0$). Therefore, for all
$(\varepsilon_{n})_{n\in\mathbb{N}}$ in $(0,\infty)$ with
$\varepsilon_{n}\downarrow 0$ as $n\rightarrow\infty$, there exists a
subsequence $(\varepsilon_{n_{m}})_{m\in\mathbb{N}}$ such that by Fatou’s
lemma

$\displaystyle\liminf_{m\rightarrow\infty}\mathbb{E}[\bm{N}_{\varepsilon_{n_{m}}}^{\\#}(Q_{L},\lambda)]\geq\mathbb{E}[\liminf_{m\rightarrow\infty}\bm{N}_{\varepsilon_{n_{m}}}^{\\#}(Q_{L},\lambda)]\geq\mathbb{E}{[\bm{N}^{\\#}(Q_{L},\lambda-\mu)]}.$

From this the inequality follows. ∎

######  Theorem 5.41.

$\bm{N}_{\varepsilon}\to\bm{N}$ vaguely.

###### Proof.

Let $Q\vcentcolon=[-1/2,1/2]^{d}$, $L\in\mathbb{N}$ and $\mu\in(0,1)$. Let
$\lambda\in\mathbb{R}$ be a continuity point of $\bm{N}$. By (54) (see also
Remark 5.34)

$\bm{N}(\lambda)-\bm{N}_{\varepsilon}(\lambda)\leq\frac{1}{L^{d}}\mathbb{E}[\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(Q_{L},\lambda)-\bm{N}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}}}(Q_{L},\lambda)].$

By Lemma 5.40, for all $L\geq 1$ and $\mu>0$,

$A\vcentcolon=\limsup_{\varepsilon\downarrow
0}\\{\bm{N}(\lambda)-\bm{N}_{\varepsilon}(\lambda)\\}\leq\frac{1}{L^{d}}\mathbb{E}[\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(Q_{L},\lambda)-\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(Q_{L},\lambda-\mu)].$

Therefore, by (54), taking the infimum over $L\geq 1$ in the above inequality,
we obtain $A\leq\bm{N}(\lambda)-\bm{N}(\lambda-\mu)$ for all $\mu>0$. As
$\lambda$ is a continuity point of $\bm{N}$, it follows that $A\leq 0$.
Similarly, one can show

$\liminf_{\varepsilon\downarrow
0}\\{\bm{N}(\lambda)-\bm{N}_{\varepsilon}(\lambda)\\}\geq 0.\qed$

######  Theorem 5.42.

One has the following tail estimates of the IDS.

  1. (a)

One has
$\lim_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\bm{N}(\lambda)=\frac{\lvert
B(0,1)\rvert}{(2\pi)^{d}}$.

  2. (b)

For every bounded domain $U$ and every $\alpha\in(0,\infty)$, one has

$\displaystyle\limsup_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\bm{N}(\lambda)$
$\displaystyle=\limsup_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\mathbb{P}(\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}(U)\leq\lambda),$
(57)
$\displaystyle\liminf_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\bm{N}(\lambda)$
$\displaystyle=\liminf_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\mathbb{P}(\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}(U)\leq\lambda).$
(58)

###### Proof.

(a) Let $Q\vcentcolon=[0,1]^{d}$. By applying Fatou’s lemma and the first
inequality of (54), we obtain

$\mathbb{E}[\liminf_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(Q,\lambda)]\leq\liminf_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\bm{N}(\lambda)\leq\limsup_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\bm{N}(\lambda).$

Since
$\lim_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(Q,\lambda)=\frac{\lvert
B(0,1)\rvert}{(2\pi)^{d}}$ by Proposition 5.29, the lower bound is obtained.
To obtain the upper bound, by the last inequality of (54) and the estimate
(52) from Lemma 5.28, for any $\theta\in(0,1)$ we have

$\displaystyle\limsup_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\bm{N}(\lambda)$
$\displaystyle\leq\limsup_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\mathbb{E}[\overline{\bm{N}}^{\scaleobj{0.8}{\mathrm{N}}}(Q,\lambda)]$
$\displaystyle\leq\limsup_{\lambda\to\infty}\lambda^{-\frac{d}{2}}(1+\theta)\frac{|B(0,1)|}{(2\pi)^{d}}\mathbb{E}[\\{\lambda+\theta+C_{Q,\theta,r}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{\xi}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert^{l}\\}^{\frac{d}{2}}]$
$\displaystyle\leq(1+\theta)\frac{|B(0,1)|}{(2\pi)^{d}}.$

Now the identity in (a) follows.

(b) Let $\lambda<0$. Thanks to the monotonicity of
$\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}$ (see Proposition 4.13), we may and
do assume $U=[0,L]^{d}$ for some $L\in(0,\infty)$. By (54) (see also Remark
5.34),

$\mathbb{P}(\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}(U)\leq\lambda)\leq\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)]\leq
L^{d}\bm{N}(\lambda).$

Therefore, we establish that the left-hand sides are greater or equal to the
right-hand sides of (57) and (58). By Lemma 4.31 (a), for $l\in(0,L/2)$ and
$n\in\mathbb{N}$, one has

$\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{n},\lambda)\leq\sum_{k\in\mathbb{Z}^{d}\cap[-1,n+1]^{d}}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(k+[-l,L+l]^{d},\lambda+Kl^{-2})$

and hence

$\frac{1}{n^{d}}\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{n},\lambda)]\leq\frac{(n+2)^{d}}{n^{d}}\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}([0,L+2l]^{d},\lambda+Kl^{-2})].$

Letting $n\to\infty$, for $p,q\in(1,\infty)$ with $p^{-1}+q^{-1}=1$, one
obtains

$\displaystyle\bm{N}(\lambda)$
$\displaystyle\leq\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}([0,L+2l]^{d},\lambda+Kl^{-2})]$
$\displaystyle=\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}([0,L+2l]^{d},\lambda+Kl^{-2})\mathbbm{1}_{\\{\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}([0,L+2l]^{d})\leq\lambda+Kl^{-2}\\}}]$
$\displaystyle\leq\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}([0,L+2l]^{d},Kl^{-2})^{q}]^{\frac{1}{q}}\mathbb{P}(\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}([0,L+2l]^{d})\leq\lambda+Kl^{-2})^{\frac{1}{p}}.$

where we applied Hölder’s inequality in the second inequality. Note that

$\mathbb{E}[\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}([0,L+2l]^{d},Kl^{-2})^{q}]<\infty$

by Lemma 4.22 and Lemma 4.23 (b). Therefore, for $U=[0,L+2l]^{d}$, the left-
hand sides are less or equal to the right-hand sides of (57) and (58). As $L$
and $l\in(0,L/2)$ can be chosen arbitrarily, the equalities follow. ∎

##  Appendix A Estimates related to function spaces

###  A.1 Estimates in Besov spaces

This subsection gives estimates in weighted Besov spaces (see Definition 2.2).

######  Lemma A.1.

Let $p,q\in[1,\infty]$, $r\in\mathbb{R}$ and
$\sigma_{1},\sigma_{2}\in[0,\infty)$. Then

$\displaystyle\lVert f\rVert_{\mathcal{C}^{r,\sigma_{1}}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{p,r,\sigma_{1}}\lVert
f\rVert_{B_{p,\infty}^{r+\frac{d}{p},\sigma_{1}}(\mathbb{R}^{d})},$ (59)
$\displaystyle\lVert f\rVert_{\mathcal{C}^{r,\sigma_{1}}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{p,q,r,\kappa,\sigma_{1}}\lVert
f\rVert_{B_{p,q}^{r+\frac{d}{p}+\kappa,\sigma_{1}}(\mathbb{R}^{d})},\qquad\kappa>0.$
(60)

If $p\sigma_{2}>d$, then

$\lVert
f\rVert_{B_{p,q}^{r,\sigma_{1}+\sigma_{2}}(\mathbb{R}^{d})}\lesssim_{p,q,r,\sigma_{1},\sigma_{2}}\lVert
f\rVert_{\mathcal{C}^{r,\sigma_{1}}(\mathbb{R}^{d})}.$ (61)

###### Proof.

By [71, Theorem 6.5], one has $\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}\sim_{p,q,r,\sigma}\lVert
w_{\sigma}f\rVert_{B_{p,q}^{r}(\mathbb{R}^{d})}$. Therefore (59) and (60)
follow by the (unweighted) Besov embedding, see [70, Section 2.7.1]. For (61),
the product estimate in the Besov space (see [51, Corollary 2.1.35], which
follows also from [59, Lemma 2.1]) yields

$\lVert
f\rVert_{B_{p,q}^{r,\sigma_{1}+\sigma_{2}}(\mathbb{R}^{d})}\lesssim_{p,q,r,\sigma_{1},\sigma_{2}}\lVert
w_{\sigma_{2}}\rVert_{B^{\lvert
r\rvert+\frac{1}{2}}_{p,q}(\mathbb{R}^{d})}\lVert
f\rVert_{\mathcal{C}^{r,\sigma_{1}}(\mathbb{R}^{d})}.$

Now $\lVert w_{\sigma_{2}}\rVert_{B^{\lvert
r\rvert+\frac{1}{2}}_{p,q}(\mathbb{R}^{d})}\lesssim\lVert
w_{\sigma_{2}}\rVert_{B^{\lvert
r\rvert+\frac{3}{4}}_{p,p}(\mathbb{R}^{d})}\lesssim\lVert
w_{\sigma_{2}}\rVert_{W^{\lvert
r\rvert+\frac{3}{4}}_{p}(\mathbb{R}^{d})}\lesssim\lVert
w_{\sigma_{2}}\rVert_{W^{\lvert r\rvert+1}_{p}(\mathbb{R}^{d})}$ (by Lemma
2.6). Since
$\lvert\partial^{m}w_{\sigma_{2}}\rvert\lesssim_{m,\sigma_{2}}w_{\sigma_{2}-\lvert
m\rvert}$, we have $\lVert w_{\sigma_{2}}\rVert_{W^{\lvert
r\rvert+1}_{p}(\mathbb{R}^{d})}<\infty$ if $p\sigma_{2}>d$. ∎

######  Theorem A.2 (Weighted Young’s inequality).

Let $p,q,r\in[1,\infty]$, $\frac{1}{r}+1=\frac{1}{p}+\frac{1}{q}$ and
$\sigma\in[0,\infty)$. Then

$\displaystyle\|w_{\sigma}(f*g)\|_{L^{r}}\lesssim_{\sigma}\|w_{-\sigma}f\|_{L^{p}}\|w_{\sigma}g\|_{L^{q}}.$

###### Proof.

Using that $w_{\sigma}(x)\lesssim_{\sigma}w_{\sigma}(x-y)w_{-\sigma}(y)$, one
can estimate $w_{\sigma}(x)|f*g|(x)\leq[(|f|w_{-\sigma})*(|g|w_{\sigma})](x)$,
see also [56, Theorem 2.4]. The rest follows by Young’s inequality, [4,
Theorem 1.4]. ∎

######  Theorem A.3.

Let $r\in\mathbb{R}$ and $\sigma\in[0,\infty]$. Let
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$ and $\int\varphi=1$ and
$\varphi_{\varepsilon}(x)=\varepsilon^{-d}\varphi(\varepsilon^{-1}x)$. Then,
for all $\eta\in\mathcal{C}^{r,\sigma}(\mathbb{R}^{d}),\delta>0$,

$\displaystyle\lVert\varphi_{\varepsilon}*\eta-\eta\rVert_{\mathcal{C}^{r-\delta,\sigma+\delta}(\mathbb{R}^{d})}\xrightarrow{\varepsilon\downarrow
0}0.$

###### Proof.

In this proof we refrain from writing “$(\mathbb{R}^{d})$”. Let $\delta>0$ and
$p\in(\frac{d}{\delta},\infty)$. By Lemma A.1, (61), $\eta$ is an element of
$B_{p,1}^{r,\sigma+\delta}$. As by Lemma A.1, (59),

$\displaystyle\lVert\varphi_{\varepsilon}*\eta-\eta\rVert_{\mathcal{C}^{r-\delta,\sigma+\delta(\mathbb{R}^{d})}}\lesssim\lVert\varphi_{\varepsilon}*\eta-\eta\rVert_{B_{p,1}^{r,\sigma+\delta}}=\sum_{j=-1}^{\infty}2^{rj}\|w_{\sigma+\delta}(\varphi_{\varepsilon}*\Delta_{j}\eta-\Delta_{j}\eta)\|_{L^{p}}.$

It suffices to show for all $i$ that
$\|w_{\sigma+\delta}(\varphi_{\varepsilon}*\Delta_{j}\eta-\Delta_{j}\eta)\|_{L^{p}}\xrightarrow{\varepsilon\downarrow
0}0$ and

$\displaystyle\|w_{\sigma+\delta}(\varphi_{\varepsilon}*\Delta_{j}\eta-\Delta_{j}\eta)\|_{L^{p}}\lesssim\|w_{\sigma+\delta}\Delta_{j}\eta\|_{L^{p}}$
(62)

As $\varphi_{\varepsilon}*\Delta_{j}\eta$ converges to $\Delta_{j}\eta$ almost
everywhere (as it does at every Lebesgue point, see [39, Proposition 2.3.8]),
the converges follows from (62) by Lebesgue’s dominated convergence theorem.
By the weighted Young inequality, we have
$\|w_{\sigma+\delta}(\varphi_{\varepsilon}*\Delta_{j}\eta)\|_{L^{p}}\lesssim_{\sigma}\|w_{-(\sigma+\delta)}\varphi_{\varepsilon}\|_{L^{1}}\|w_{\sigma+\delta}\Delta_{j}\eta\|_{L^{p}}$.
As $w_{-(\sigma+\delta)}(\varepsilon x)\leq w_{-(\sigma+\delta)}(x)$ for
$\varepsilon\in(0,1)$, we have
$\|w_{-(\sigma+\delta)}\varphi_{\varepsilon}\|_{L^{1}}\leq\|w_{-(\sigma+\delta)}\varphi\|_{L^{1}}$
which is finite because $\varphi\in\mathcal{S}(\mathbb{R}^{d})$. This proves
(62). ∎

######  Lemma A.4.

Let $p,q\in[1,\infty]$, $r\in\mathbb{R}$ and $\sigma\in[0,\infty)$. Let $Z\in
B_{p,q}^{r,\sigma}(\mathbb{R}^{d})$ and $\phi\in\mathcal{S}(\mathbb{R}^{d})$.
Then, one has

$\lVert\phi(L^{-1}\cdot)Z\rVert_{B^{r}_{p,q}(\mathbb{R}^{d})}\lesssim_{p,q,r,\sigma,\phi}L^{\sigma}\lVert
Z\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})},\qquad L\geq 1.$

###### Proof.

By the product estimate in the Besov space [63, Theorem 4.37], we have

$\lVert\phi(L^{-1}\cdot)Z\rVert_{B_{p,q}^{r}(\mathbb{R}^{d})}\lesssim_{p,q,r}\lVert\phi(L^{-1}\cdot)w_{-\sigma}\rVert_{\mathcal{C}^{\lvert
r\rvert+\frac{1}{2}}(\mathbb{R}^{d})}\lVert
w_{\sigma}Z\rVert_{B_{p,q}^{r}(\mathbb{R}^{d})}.$

Since $\lVert
w_{\sigma}Z\rVert_{B_{p,q}^{r}(\mathbb{R}^{d})}\lesssim_{p,q,r,\sigma}\lVert
Z\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}$ by [71, Theorem 6.5] and
$\lVert\cdot\rVert_{\mathcal{C}^{|r|+\frac{1}{2}}}\lesssim\lVert\cdot\rVert_{W_{\infty}^{|r|+\frac{1}{2}}}\lesssim\lVert\cdot\rVert_{W_{\infty}^{|r|+1}}$
by Lemma 2.6, it suffices to show

$\lVert\phi(L^{-1}\cdot)w_{-\sigma}\rVert_{W_{\infty}^{\lvert
r\rvert+1}(\mathbb{R}^{d})}\lesssim_{r,\sigma,\phi}L^{\sigma}.$

For this, it suffices to show

$\lVert\partial^{m}[w_{-\sigma}\phi(L^{-1}\cdot)]\rVert_{L^{\infty}(\mathbb{R}^{d})}\lesssim_{\sigma,m,\phi}L^{\sigma}$

for every $m\in\mathbb{N}_{0}^{d}$. By the Leibniz rule,

$\partial^{m}[w_{-\sigma}\phi(L^{-1}\cdot)]=\sum_{l=0}^{m}\binom{m}{l}L^{-\lvert
m-l\rvert}\partial^{l}w_{-\sigma}\partial^{m-l}\phi(L^{-1}\cdot).$

Since $\lvert\partial^{l}w_{-\sigma}(x)\rvert\lesssim_{\sigma,l}(1+\lvert
x\rvert^{2})^{\frac{\sigma-\lvert l\rvert}{2}}$, we obtain

$\displaystyle\sup_{x\in\mathbb{R}^{d}}\lvert\partial^{m}[w_{-\sigma}\phi(L^{-1}\cdot)](x)\rvert$
$\displaystyle\lesssim_{\sigma,m}\sum_{l=0}^{m}\binom{m}{l}L^{-\lvert
m-l\rvert}\sup_{x\in\mathbb{R}^{d}}(1+\lvert x\rvert^{2})^{\frac{\sigma-\lvert
l\rvert}{2}}\lvert\partial^{m-l}\phi(L^{-1}x)\rvert$
$\displaystyle=\sum_{l=0}^{m}\binom{m}{l}L^{-\lvert
m-l\rvert}\sup_{x\in\mathbb{R}^{d}}(1+\lvert
Lx\rvert^{2})^{\frac{\sigma-\lvert
l\rvert}{2}}\lvert\partial^{m-l}\phi(x)\rvert$
$\displaystyle\lesssim_{m,\sigma,\phi}L^{\sigma}.\qed$

######  Lemma A.5.

Let $U$ be a bounded domain, $r\in\mathbb{R}$ and $\sigma\in(0,\infty)$. Then
for for $Z\in\mathcal{C}^{r,\sigma}(\mathbb{R}^{d})$

$\displaystyle\lVert Z\rVert_{C^{r}(U_{L})}\lesssim_{U,\sigma}L^{\sigma}\lVert
Z\rVert_{\mathcal{C}^{r,\sigma}(\mathbb{R}^{d})},\qquad L\geq 1.$

###### Proof.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ be $1$ on a neighborhood of $U$.
Then $\lVert
Z\rVert_{C^{r}(U_{L})}=\lVert\phi(L\cdot)Z\rVert_{C^{r}(U)}\leq\lVert\phi(L\cdot)Z\rVert_{C^{r}(\mathbb{R}^{d})}$.
By Lemma 2.6,

$\displaystyle\lVert\phi(L\cdot)Z\rVert_{C^{r}(\mathbb{R}^{d})}\lesssim_{r}\lVert\phi(L\cdot)Z\rVert_{\mathcal{C}^{r}(\mathbb{R}^{d})}.$

Therefore, we obtain the desired estimate by an application of Lemma A.4. ∎

######  Lemma A.6.

Let $p,q\in[1,\infty]$, $r\in\mathbb{R}$, $\sigma\in[0,\infty)$,
$m\in\mathbb{N}_{0}^{d}$ and $a\in\mathbb{R}$.

  1. (a)

One has $\lVert\partial^{m}f\rVert_{B^{r-\lvert
m\rvert,\sigma}_{p,q}(\mathbb{R}^{d})}\lesssim_{p,q,r,\sigma,m}\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}$.

  2. (b)

Let $\tilde{\chi}$ be a smooth function on $\mathbb{R}^{d}$ such that
$\tilde{\chi}=0$ in a neighborhood of $0$ and all the derivatives are bounded.
Then, one has $\lVert\mathcal{F}^{-1}[\lvert
2\pi\cdot\rvert^{2a}\tilde{\chi}\mathcal{F}f]\rVert_{B_{p,q}^{r-2a,\sigma}(\mathbb{R}^{d})}\lesssim_{p,q,r,\sigma,a,\tilde{\chi}}\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}$.

###### Proof.

We only prove (a), as the proof of (b) is similar. We use the notations from
Definition 2.2. Let $\psi$ be a compactly supported smooth function on
$\mathbb{R}^{d}$ such that $\psi=1$ on a neighborhood of
$\operatorname{supp}(\chi)$ and set $\psi_{j}\vcentcolon=\psi(2^{-j}\cdot)$.
Because
$\Delta_{j}\partial^{m}f=\mathcal{F}^{-1}[\chi(2^{-j}\cdot)\psi_{j}]*\Delta_{j}f$
for $j\in\mathbb{N}_{0}$, by Theorem A.2 we have

$\lVert
w_{\sigma}(\Delta_{j}\partial^{m}f)\rVert_{L^{p}(\mathbb{R}^{d})}\leq\lVert
w_{-\sigma}\mathcal{F}^{-1}[(-2\pi
i\cdot)^{m}\psi_{j}]\rVert_{L^{1}(\mathbb{R}^{d})}\lVert
w_{\sigma}\Delta_{j}f\rVert_{L^{p}(\mathbb{R}^{d})}.$ (63)

It remains to observe that for all $j\in\mathbb{N}_{0}$

$\displaystyle 2^{-j\lvert m\rvert}\lVert w_{-\sigma}\mathcal{F}^{-1}[(-2\pi
i\cdot)^{m}\psi_{j}]\rVert_{L^{1}(\mathbb{R}^{d})}$ $\displaystyle=\lVert
w_{-\sigma}\mathcal{F}^{-1}[(-2^{-j+1}\pi
i\cdot)^{m}\psi(2^{-j}\cdot)]\rVert_{L^{1}(\mathbb{R}^{d})}$
$\displaystyle=\lVert
2^{jd}\Big{[}w_{-\sigma}(2^{-j}\cdot)[\mathcal{F}^{-1}[(-2\pi
i\cdot)^{m}\psi\Big{]}(2^{j}\cdot)\rVert_{L^{1}(\mathbb{R}^{d})}$
$\displaystyle\leq\lVert w_{-\sigma}[\mathcal{F}^{-1}[(-2\pi
i\cdot)^{m}\psi]\rVert_{L^{1}(\mathbb{R}^{d})},$

as $w_{-\sigma}(2^{-j}\cdot)\leq w_{-\sigma}$. For $j=-1$ a similar estimate
as (63) holds for a $\tilde{\psi}\in C_{c}^{\infty}(\mathbb{R}^{d})$ with
$\tilde{\psi}=1$ on $\operatorname{supp}(\tilde{\chi})$. ∎

######  Definition A.7.

For $J\in\mathbb{N}_{0}$ or $J=-1$ we write

$\displaystyle\Delta_{\leq J}f=\sum_{j=-1}^{J}\Delta_{j}f,\qquad\Delta_{\geq
J}f=\sum_{j=J}^{\infty}\Delta_{j}f.$

######  Remark A.8.

Observe that by definition of $\check{\chi}$ and $\chi$ (Definition 2.2), for
$N\in\mathbb{N}$

$\displaystyle(1-\check{\chi})(2^{-N}x)=\sum_{j=N}^{\infty}\chi(2^{-j}x),\qquad
x\in\mathbb{R}^{d},$

and therefore

$\displaystyle\Delta_{\geq N}f$
$\displaystyle=\mathcal{F}^{-1}\Big{(}(1-\check{\chi})(2^{-N}\cdot)\mathcal{F}f\Big{)},\quad\Delta_{\leq
N}f=\mathcal{F}^{-1}\Big{(}\check{\chi}(2^{-N}\cdot)\mathcal{F}f\Big{)}.$

######  Lemma A.9.

Let $p,q\in[1,\infty]$, $r,s\in\mathbb{R}$ with $r\leq s$,
$\sigma\in[0,\infty)$ and $N\in\mathbb{N}_{0}$. Then, one has (observe the
difference of the positions of $r$ and $s$)

$\displaystyle\lVert\Delta_{\geq
N}f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{s-r,\sigma}2^{-(s-r)N}\lVert
f\rVert_{B_{p,q}^{s,\sigma}(\mathbb{R}^{d})},$
$\displaystyle\lVert\Delta_{\leq
N}f\rVert_{B_{p,q}^{s,\sigma}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{s-r,\sigma}2^{(s-r)N}\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}.$

###### Proof.

We first observe by Remark A.8 that $\Delta_{\geq
N}\Delta_{j}f=[2^{Nd}\mathcal{F}^{-1}(1-\check{\chi})(2^{N}\cdot)*f$. Thus, by
[56, Theorem 2.4 and Lemma 2.6], one has

$\lVert w_{\sigma}(\Delta_{j}\Delta_{\geq
N}f)\rVert_{L^{p}(\mathbb{R}^{d})}\lesssim\lVert
w_{\sigma}(\Delta_{j}f)\rVert_{L^{p}(\mathbb{R}^{d})}$

Therefore,

$\displaystyle\lVert\Delta_{\geq
N}f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}$
$\displaystyle=\Big{(}\sum_{j=N-1}^{\infty}2^{jqr}\lVert
w_{\sigma}(\Delta_{j}\Delta_{\geq
N}f)\rVert_{L^{p}(\mathbb{R}^{d})}^{q}\Big{)}^{\frac{1}{q}}$
$\displaystyle\leq 2^{-(N-1)(s-r)}\Big{(}\sum_{j=N-1}^{\infty}2^{jqs}\lVert
w_{\sigma}(\Delta_{j}\Delta_{\geq
N}f)\rVert_{L^{p}(\mathbb{R}^{d})}^{q}\Big{)}^{\frac{1}{q}}$
$\displaystyle\lesssim_{s-r,\sigma}2^{-N(s-r)}\Big{(}\sum_{j=N-1}^{\infty}2^{jqs}\lVert
w_{\sigma}(\Delta_{j}f)\rVert_{L^{p}(\mathbb{R}^{d})}^{q}\Big{)}^{\frac{1}{q}}$
$\displaystyle\leq 2^{-N(s-r)}\lVert
f\rVert_{B_{p,q}^{s,\sigma}(\mathbb{R}^{d})}.$

The second inequality can be proven similarly. ∎

Recall the definition of an admissible kernel and of $G_{N}$, see Definition
3.4 and Definition 3.1.

######  Corollary A.10.

Let $p,q\in[1,\infty]$, $r,s\in\mathbb{R}$ with $r\leq s$,
$\sigma\in[0,\infty)$ and $N\in\mathbb{N}_{0}$. Let $K$ be an admissible
kernel. Set $H_{N}\vcentcolon=G_{N}-K$. Then, one has

$\displaystyle\lVert G_{N}\ast f\rVert_{B_{p,q}^{r+2,\sigma}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{p,q,r,s,\sigma}2^{-(s-r)N}\lVert
f\rVert_{B_{p,q}^{s,\sigma}(\mathbb{R}^{d})},$ $\displaystyle\lVert H_{N}\ast
f\rVert_{B_{p,q}^{s+2,\sigma}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{p,q,r,s,\sigma}2^{(s-r)N}\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})},$
$\displaystyle\lVert(G_{N}-G_{0})\ast
f\rVert_{B_{p,q}^{s+2,\sigma}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{p,q,r,s,\sigma}2^{(s-r)N}\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}.$

###### Proof.

Suppose $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ is $0$ in a neighborhood of
$0$ and is equal to $1$ on $\operatorname{supp}(1-\check{\chi})$. If we set
$g\vcentcolon=\mathcal{F}^{-1}[\lvert 2\pi\cdot\rvert^{-2}\psi\mathcal{F}f]$,
then $G_{N}\ast f=\Delta_{\geq N}g$. Therefore, the first claimed inequality
follows from Lemma A.6 (b) and Lemma A.9.

To prove the second claimed inequality, recall that one has
$H_{N}=(G_{N}-G_{0})+(G_{0}-K)$. By Lemma C.4 below, $G_{0}-K$ belongs to
$\mathcal{S}(\mathbb{R}^{d})$. Therefore,

$\lVert(G_{0}-K)\ast
f\rVert_{B_{p,q}^{s+2,\sigma}(\mathbb{R}^{d})}\lesssim_{p,q,r,s,\sigma}\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}.$

On the other hand, one has $(G_{N}-G_{0})\ast f=(\Delta_{\geq N}-\Delta_{\geq
0})g=(\Delta_{\leq N-1}-\Delta_{-1})g$. Hence, by Lemma A.6 (b) and by Lemma
A.9, the third inequality and thus the second follow. ∎

###  A.2 Estimates of constants of functional inequalities on bounded domains

In Definition 4.10 we have introduced the smallest constant that appears in
interpolation inequalities. In this section we introduce also other constants
that appear in functional inequalities and study their behaviour (also under
scaling of the underlying domain).

######  Definition A.11.

Let $U$ be a bounded domain and $p,p_{1},p_{2}\in[1,\infty]$,
$r_{1},r_{2},s\in[0,\infty)$, $r\in(0,\infty)$ and $\delta\in(0,r)$. We set

$\displaystyle C^{U}_{\operatorname{Embed}}[W^{r_{1}}_{p_{1}}\to
W^{r_{2}}_{p_{2}}]\vcentcolon=\sup_{f\in
W^{r_{1}}_{p_{1}}(U)\setminus\\{0\\}}\frac{\lVert
f\rVert_{W^{r_{2}}_{p_{2}}(U)}}{\lVert f\rVert_{W^{r_{1}}_{p_{1}}(U)}},$
$\displaystyle C^{U}_{\operatorname{Prod}}[W_{2p}^{r}\to
W_{p}^{r-\delta}]\vcentcolon=\sup_{f\in
W^{r}_{2p}(U)\setminus\\{0\\}}\frac{\lVert
f^{2}\rVert_{W^{r-\delta}_{p}(U)}}{\lVert f\rVert_{W^{r}_{2p}(U)}^{2}}.$

Similarly, we set $C^{U}_{\operatorname{Embed}}[W^{r_{1}}_{p_{1},0}\to
W^{r_{2}}_{p_{2},0}],\ldots$ by replacing the function spaces “$W_{p}^{r}$” to
those with zero boundary conditions “$W_{p,0}^{r}$”. If $U$ is a bounded
Lipschitz domain, for a universal extension operator $\iota$ from $U$ to
$\mathbb{R}^{d}$ as in Lemma 2.7, we set

$\displaystyle
C^{U}_{\operatorname{Ext}}[W^{r_{1}}_{p_{1}},W^{r_{2}}_{p_{2}}]\vcentcolon=\inf\\{\lVert\iota\rVert_{W^{r_{1}}_{p_{1}}(U)\rightarrow
W^{r_{1}}_{p_{1}}(\mathbb{R}^{d})}+\lVert\iota\rVert_{W^{r_{2}}_{p_{2}}(U)\rightarrow
W^{r_{2}}_{p_{2}}(\mathbb{R}^{d})}$ $\displaystyle\hskip
142.26378pt|\iota\mbox{ is an universal extension operator}\\},$
$\displaystyle C^{\partial
U}_{\operatorname{R}}(W^{r}_{p})\vcentcolon=\inf\\{\lVert\mathcal{R}\rVert_{W^{r}_{p}(\partial
U)\to W^{r+\frac{1}{p}}_{p}(U)}\ |\ \mathcal{R}\mbox{ is a right inverse of
}\mathcal{T}_{W_{p}^{r+\frac{1}{p}}(U)}\\},$ $\displaystyle C^{\partial
U}_{\operatorname{Prod}}[W_{2p}^{r}\to W_{p}^{r-\delta}]\vcentcolon=\sup_{f\in
W^{r}_{2p}(\partial U)\setminus\\{0\\}}\frac{\lVert
f^{2}\rVert_{W^{r-\delta}_{p}(\partial U)}}{\lVert
f\rVert_{W^{r}_{2p}(\partial U)}^{2}},$ $\displaystyle
C^{U}_{\operatorname{Mult}}[W_{p}^{r}]\vcentcolon=\sup_{f\in
W_{p}^{r}(U)\setminus\\{0\\}}\frac{\lVert\mathbbm{1}_{U}f\rVert_{W_{p}^{r}(\mathbb{R}^{d})}}{\lVert
f\rVert_{W^{r}_{p}(U)}},$

and
$C^{U}_{\operatorname{Ext}}[W^{r}_{p}]\vcentcolon=C^{U}_{\operatorname{Ext}}[W^{r}_{p},W^{r}_{p}]$.

######  Lemma A.12.

Let $U$ be a domain.

  1. (a)

Let $p_{1},p_{2}\in(1,\infty)$ with $p_{1}\leq p_{2}$ and
$r_{1},r_{2}\in[0,\infty)$ with
$r_{2}=r_{1}-d(\frac{1}{p_{1}}-\frac{1}{p_{2}})$. Then, one has
$C^{U}_{\operatorname{Embed}}[W^{r_{1}}_{p_{1},0}\to
W^{r_{2}}_{p_{2},0}]\lesssim_{p_{1},p_{2},r_{1}}1$. If $U$ is a bounded
Lipschitz domain, one has $C^{U}_{\operatorname{Embed}}[W^{r_{1}}_{p_{1}}\to
W^{r_{2}}_{p_{2}}]\lesssim_{p_{1},p_{2},r_{1}}C^{U}_{\operatorname{Ext}}[W^{r_{1}}_{p_{1}}]$.

  2. (b)

Let $s\in(0,1)$. Then, one has
$C^{U}_{\operatorname{IP}}[H^{s}_{0}]\lesssim_{s}1$. If $U$ is a bounded
Lipschitz domain, one has
$C^{U}_{\operatorname{IP}}[H^{s}]\lesssim_{s}C^{U}_{\operatorname{Ext}}[L^{2},H^{1}]$.

###### Proof.

We only prove the claim for a bounded Lipschitz domain $U$.

(a) Let $\iota$ be a universal extension operator from $U$ to
$\mathbb{R}^{d}$. Then, by using the Sobolev embedding in $\mathbb{R}^{d}$
[11, Theorem 8.12.6] for the second inequality,

$\lVert
f\rVert_{W^{r_{2}}_{p_{2}}(U)}\leq\lVert\iota(f)\rVert_{W^{r_{2}}_{p_{2}}(\mathbb{R}^{d})}\lesssim_{p_{1},p_{2},r_{1}}\lVert\iota(f)\rVert_{W^{r_{1}}_{p_{1}}(\mathbb{R}^{d})}\leq\lVert\iota\rVert_{W_{p_{1}}^{r_{1}}(U)\rightarrow
W_{p_{1}}^{r_{1}}(\mathbb{R}^{d})}\lVert f\rVert_{W^{r_{1}}_{p_{1}}(U)},$

and thus $\lVert
f\rVert_{W^{r_{2}}_{p_{2}}(U)}\lesssim_{p_{1},p_{2},r_{1}}C^{U}_{\operatorname{Extend}}[W^{r_{1}}_{p_{1}}]\lVert
f\rVert_{W^{r_{1}}_{p_{1}}(U)}$.

(b) We can prove the claim similarly by using the inequality [4, Proposition
2.22]

$\lVert f\rVert_{H^{s}(\mathbb{R}^{d})}\lesssim_{s}\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{1-s}\lVert
f\rVert_{H^{1}(\mathbb{R}^{d})}^{s}.\qed$

######  Lemma A.13.

Let $U$ be a bounded domain, $p\in[1,\infty)$, $r\in(0,1)$ and
$\varepsilon\in(0,r)$. Then we have

$C^{U}_{\operatorname{Prod}}[W_{2p,0}^{r}\to
W_{p,0}^{r-\varepsilon}]\lesssim_{p,\varepsilon}1,$

and if $U$ is a bounded Lipschitz domain

$C^{U}_{\operatorname{Prod}}[W_{2p}^{r}\to
W_{p}^{r-\varepsilon}]\lesssim_{p,\varepsilon}1,$ $C^{\partial
U}_{\operatorname{Prod}}[W_{2p}^{r}\to
W_{p}^{r-\varepsilon}]\lesssim_{p,\varepsilon}1+\sup_{x\in\partial
U}\Big{(}\int_{\partial U}\frac{\,\mathrm{d}y}{\lvert
x-y\rvert^{d-1-2p\varepsilon}}\Big{)}^{\frac{1}{2p}}.$

###### Proof.

We only prove the first inequality. Since

$\int_{\lvert x\rvert\geq 1}\frac{1}{\lvert
x\rvert^{d+p(r-\varepsilon)}}\,\mathrm{d}x<\infty,$

one has

$\lVert
f^{2}\rVert_{W_{p}^{r-\varepsilon}(\mathbb{R}^{d})}\lesssim_{p,r}\lVert
f^{2}\rVert_{L^{p}(\mathbb{R}^{d})}+\int_{\lvert x-y\rvert\leq 1}\frac{\lvert
f(x)^{2}-f(y)^{2}\rvert^{p}}{\lvert
x-y\rvert^{d+p(r-\varepsilon)}}\,\mathrm{d}x\,\mathrm{d}y.$

Observe that $\lVert f^{2}\rVert_{L^{p}(\mathbb{R}^{d})}\leq\lVert
f\rVert^{2}_{L^{2p}(\mathbb{R}^{d})}$. Furthermore, observe that

$\frac{\lvert f(x)^{2}-f(y)^{2}\rvert^{p}}{\lvert
x-y\rvert^{d+p(r-\varepsilon)}}=\Big{(}\frac{\lvert f(x)+f(y)\rvert}{\lvert
x-y\rvert^{\frac{d}{2p}-\varepsilon}}\Big{)}^{p}\Big{(}\frac{\lvert
f(x)-f(y)\rvert}{\lvert x-y\rvert^{\frac{d}{2p}+r}}\Big{)}^{p},$

so that, by Hölder’s inequality

$\int_{\lvert x-y\rvert\leq 1}\frac{\lvert f(x)^{2}-f(y)^{2}\rvert^{p}}{\lvert
x-y\rvert^{d+p(r-\varepsilon)}}\,\mathrm{d}x\,\mathrm{d}y\leq\lVert
f\rVert_{W^{r}_{2p}(\mathbb{R}^{d})}\Big{(}\int_{\lvert x-y\rvert\leq
1}\frac{\lvert f(x)+f(y)\rvert^{2p}}{\lvert
x-y\rvert^{d-2p\varepsilon}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{\frac{1}{2p}}.$

Now the latter integral can be estimated by $\|f\|_{L^{2p}}$ times the
following integral over the unit ball that can be estimated as follows

$\int_{\lvert x\rvert\leq 1}\frac{\,\mathrm{d}x}{\lvert
x\rvert^{d-2p\varepsilon}}\lesssim\int_{0}^{1}\frac{dr}{r^{1-2p\varepsilon}}=\frac{1}{2p\varepsilon}.\qed$

######  Lemma A.14 ([68, Proposition 5.3]).

Let $p\in(1,\infty)$, $r\in(0,\frac{1}{p})$ and let $U$ be a bounded Lipschitz
domain. Then, the map

$B_{p,p}^{r}(\mathbb{R}^{d})\rightarrow B_{p,p}^{r}(\mathbb{R}^{d}),\quad
f\mapsto f\mathbbm{1}_{U}$

is a bounded linear operator.

######  Lemma A.15.

Let $U$ be a bounded Lipschitz domain. Then, we have

$\displaystyle\sup_{L\geq
1}C^{U_{L}}_{\operatorname{Ext}}[W^{r_{1}}_{p_{1}},W^{r_{2}}_{p_{2}}]<\infty,$
$\displaystyle\quad p_{1},p_{2}\in[1,\infty],r_{1},r_{2}\in[0,\infty),$ (64)
$\displaystyle\sup_{L\geq
1}\lVert\mathcal{T}_{W^{r}_{p}(U_{L})}\rVert<\infty,$ $\displaystyle\quad
p\in(1,\infty),r\in(\tfrac{1}{p},1+\tfrac{1}{p}),$ (65)
$\displaystyle\sup_{L\geq 1}C^{\partial
U_{L}}_{\operatorname{R}}[W^{r}_{p}]<\infty,$ $\displaystyle\quad
p\in(1,\infty),r\in(0,1),$ (66) $\displaystyle\sup_{L\geq
1}C^{U_{L}}_{\operatorname{Embed}}[W^{r_{1}}_{p_{1}}\to
W^{r_{2}}_{p_{2}}]<\infty,$
$\displaystyle\quad\begin{cases}p_{1},p_{2}\in(1,\infty),r_{1},r_{2}\in[0,\infty),&\\\
p_{1}\leq p_{2},r_{2}=r_{1}-d(\frac{1}{p_{1}}-\frac{1}{p_{2}}),\end{cases}$
(67) $\displaystyle\sup_{L\geq 1}C^{U_{L}}_{\operatorname{IP}}[H^{s}]<\infty$
$\displaystyle\quad s\in(0,1),$ (68) $\displaystyle\sup_{L\geq
1}C^{U_{L}}_{\operatorname{Mult}}[W_{p}^{r}]<\infty,$ $\displaystyle\quad
p\in(1,\infty),r\in(0,\tfrac{1}{p}),$ (69) $\displaystyle\sup_{L\geq
1}C^{U_{L}}_{\operatorname{Prod}}[W_{2p}^{r}\to W_{p}^{r-\varepsilon}]<\infty$
$\displaystyle\quad p\in[1,\infty),r\in(0,1),\varepsilon\in(0,r),$ (70)
$\displaystyle\sup_{L\geq 1}L^{-\varepsilon}C^{\partial
U_{L}}_{\operatorname{Prod}}[W_{2p}^{r}\to W_{p}^{r-\varepsilon}]<\infty,$
$\displaystyle\quad p\in[1,\infty),r\in(0,1),\varepsilon\in(0,r).$ (71)

If $U$ is a bounded domain (that is not necessarily Lipschitz), then (67) and
(70) hold by replacing the occurrences of the form “$W^{a}_{b}$” by
“$W^{a}_{b,0}$”.

###### Proof.

Let $\iota$ be a universal extension operator from $U$ to $\mathbb{R}^{d}$
(Definition 2.8). For $L\geq 1$, we define a universal extension operator
$\iota_{L}$ from $U_{L}$ to $\mathbb{R}^{d}$ by
$\iota_{L}(f)\vcentcolon=\iota(f(L\cdot))(L^{-1}\cdot)$. By change of
variables, and using that
$\partial^{\alpha}\iota(f)=\iota(\partial^{\alpha}f)$ it is straightforward to
check that $\lVert\iota_{L}\rVert_{W^{r}_{p}(U_{L})\rightarrow
W^{r}_{p}(\mathbb{R}^{d})}\leq\lVert\iota\rVert_{W^{r}_{p}(U)\rightarrow
W^{r}_{p}(\mathbb{R}^{d})}$. This implies (64). The two (65) and (66)
estimates can be proven similarly.

(67) and (68) follow by Lemma A.12 and by (64).

(69) First observe $r\in(0,1)$. Set $F\vcentcolon=\iota_{L}(f)$ for $f\in
W_{p}^{r}(U_{L})$. Then, $\mathbbm{1}_{U_{L}}f=\mathbbm{1}_{U_{L}}F$ and thus
$\lVert\mathbbm{1}_{U_{L}}f\rVert_{W_{p}^{r}(\mathbb{R}^{d})}\leq\lVert
F\rVert_{L^{p}}+[\mathbbm{1}_{U_{L}}F]_{W^{r}_{p}(\mathbb{R}^{d})}$. By change
of variables, one has
$[g(L^{-1}\cdot)]_{W_{p}^{s}(\mathbb{R}^{d})}=L^{\frac{d}{p}-s}[g]_{W_{p}^{s}(\mathbb{R}^{d})}$
and thus

$[\mathbbm{1}_{U_{L}}F]_{W^{r}_{p}(\mathbb{R}^{d})}=L^{\frac{d}{p}-r}[\mathbbm{1}_{U}F(L\cdot)]_{W^{r}_{p}(\mathbb{R}^{d})}.$

By Lemma A.14 and Lemma 2.6, one has

$\displaystyle[\mathbbm{1}_{U}F(L\cdot)]_{W^{r}_{p}(\mathbb{R}^{d})}\leq\lVert\mathbbm{1}_{U}F(L\cdot)\rVert_{W^{r}_{p}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{U,p,r}\lVert
F(L\cdot)\rVert_{L^{p}(\mathbb{R}^{d})}+[F(L\cdot)]_{W^{r}_{p}(\mathbb{R}^{d})}$
$\displaystyle=L^{-\frac{d}{p}}\lVert
F\rVert_{L^{p}(\mathbb{R}^{d})}+L^{-\frac{d}{p}-r}[F]_{W^{r}_{p}(\mathbb{R}^{d})}.$

The claim follows because $\lVert
F\rVert_{W^{p}_{r}(\mathbb{R}^{d})}\leq\lVert\iota_{L}\rVert_{W_{p}^{r}(U_{L})}\lVert
f\rVert_{W^{p}_{r}(U_{L})}\leq\lVert\iota\rVert_{W_{p}^{r}(U)}\lVert
f\rVert_{W^{p}_{r}(U_{L})}$.

(70) and (71) follow from Lemma A.13. ∎

##  Appendix B A regularity structure for the gPAM

In this appendix, we consider a regularity structure for the generalized
Parabolic Anderson model

$\partial_{t}u=\Delta
u+\sum_{i,j=1}^{d}g_{i,j}(u)\partial_{i}u\partial_{j}u+\sum_{i=1}^{d}h_{i}(u)\partial_{i}u+k(u)+f(u)\xi,$

based on the abstract theory by Bruned, Hairer and Zambotti [13]. See [6] as
well.

###  B.1 Terminologies

Here we review some terminologies from [13].

######  Definition B.1.

We fix a type set $\vcentcolon=\\{\Xi,\mathscr{I}\\}$. The symbol represents
the noise $\xi$ and the symbol represents an abstract integration operator.

######  Definition B.2.

We define the following notions regarding graphs.

  1. (a)

A _rooted tree_ is a finite connected simple graph without cycles, with a
distinguished vertex called the _root_. We do not allow for an empty tree but
we allow for a _trivial tree_ $\bullet$ which consists of only one vertex.
Vertices will be called _nodes_. Given a rooted tree $T$, the set of nodes and
that of edges are denoted by $N=N_{T}$ and by $E=E_{T}$ respectively. We
denote by the root of $T$. Nodes of a rooted tree are endowed with a partial
order $\leq$ by their distances from the root. We orient edges $(x,y)\in E$ so
that $x\leq y$.

  2. (b)

A _forest_ is a finite simple graph without cycles. We say a forest is
_rooted_ if every component of the forest is a rooted tree. We allow for an
empty rooted forest. Given a rooted forest $F$, the set of nodes and that of
edges are denoted by $N=$ and by $E=$ respectively.

  3. (c)

A tree or a forest is called typed if it is endowed with a map
$:E\to\mathfrak{L}$ where $E$ is the set of edges.

  4. (d)

We say $A$ is a _subforest_ of a forest $F$, and write $A\subseteq F$, if
$N_{A}\subseteq N_{F}$ and $E_{A}\subseteq E_{F}$ and if $(x,y)\in E_{A}$
implies $\\{x,y\\}\subseteq N_{A}$. We note that a _subtree_ of a rooted tree
is again a rooted tree whose root is the unique vertex which is closest to the
root of the original rooted tree. Therefore, a subforest of a rooted forest is
again a rooted forest. If a forest is typed, a subforest inherits types by
restriction. If $A$ and $B$ are (rooted, typed) forests, we denote by $A\sqcup
B$ the disjoint union of $A$ and $B$ with types naturally inherited.

######  Definition B.3.

In this paper, a typed forest $F$ is often equipped with a _colouring_
$\hat{F}$ and _decorations_
$\scaleobj{0.8}{\mathrm{N}},\mathfrak{o},\mathfrak{e}$ as follows.

  1. (a)

A pair is called a _colourful forest_ if the following hold:

     * •

$F=(E_{F},N_{F},\mathfrak{t})$ is a typed rooted forest.

     * •

One has $\hat{F}:E_{F}\sqcup N_{F}\to\\{0,1,2\\}$ such that if
$\hat{F}((x,y))=i>0$ for $(x,y)\in E_{F}$ then $\hat{F}(x)=\hat{F}(y)=i$.

  2. (b)

If $(F,\hat{F})$ is a colourful forest and

     * •

$\scaleobj{0.8}{\mathrm{N}}:N_{F}\to\mathbb{N}_{0}^{d}$,

     * •

$\mathfrak{o}:N_{F}\to\mathbb{Z}\oplus\mathbb{Z}[\mathfrak{L}]$ with
$\operatorname{supp}(\mathfrak{o})\subseteq\cup_{i>0}\hat{F}^{-1}(i)$,

     * •

$\mathfrak{e}:E_{F}\to\mathbb{N}_{0}^{d}$ and
$\operatorname{supp}(\mathfrak{e})\subseteq
E_{F}\setminus\operatorname{supp}(\hat{F})$,

then the $5$-tuple , also written
$(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}$, is
called a _decorated forest_. We denote by $\mathfrak{F}$ the set of decorated
forests.

  3. (c)

For $x,y\in N_{F}$, we write $x\sim y$ if they are connected in
$\cup_{i>0}\hat{F}^{-1}(i)$.

  4. (d)

Given a decoreted forest
$(F,\hat{F},\scaleobj{0.8}{\mathrm{N}},\mathfrak{o},\mathfrak{e})$, we view a
subforest $A\subseteq F$ as a decorated forest by restricting the associated
maps $(\hat{F},\scaleobj{0.8}{\mathrm{N}},\mathfrak{o},\mathfrak{e})$.

  5. (e)

We write for the decorated tree $(\bullet,2,m,0,0)$.

Many examples of colourful forests can be found in [13, Section 2.2].

######  Definition B.4.

Two notions of product for forests are defined as follows.

  1. (a)

For decorated forests
$\tau_{i}=(F_{i},\hat{F}_{i},\scaleobj{0.8}{\mathrm{N}}_{i},\mathfrak{o}_{i},\mathfrak{e}_{i})$
($i=1,2$), we define the _forest product_ by

$\vcentcolon=(F_{1}\sqcup
F_{2},\hat{F}_{1}+\hat{F}_{2},\scaleobj{0.8}{\mathrm{N}}_{1}+\scaleobj{0.8}{\mathrm{N}}_{2},\mathfrak{o}_{1}+\mathfrak{o}_{2},\mathfrak{e}_{1}+\mathfrak{e}_{2})$

where, for $i\neq j$,
$(\hat{F}_{i},\scaleobj{0.8}{\mathrm{N}}_{i},\mathfrak{o}_{i},\mathfrak{e}_{i})$
are set to $0$ on $F_{j}$.

  2. (b)

For a decorated forest
$\tau=(F,\hat{F},\scaleobj{0.8}{\mathrm{N}},\mathfrak{o},\mathfrak{e})$, we
denote by $(\tau)$ the decorated tree

$(\mathscr{J}(F),[\hat{F}],[\scaleobj{0.8}{\mathrm{N}}],[\mathfrak{o}],\mathfrak{e}),$

where $\mathscr{J}(F)$ is the tree obtained by gluing all the roots of $F$,

$[\hat{F}](\rho_{\mathscr{J}(F)})\vcentcolon=\max_{y\text{ is a root of
$F$}}\hat{F}(y),\quad[\hat{F}](x)=\hat{F}(x)\,\,\mbox{for
}x\neq\rho_{\mathscr{J}(F)}$

and $[\scaleobj{0.8}{\mathrm{N}}]$ and $[\mathfrak{o}]$ are defined at the new
root by summing the values at the roots of $F$, and are equal to
$\scaleobj{0.8}{\mathrm{N}}$ and $\mathfrak{o}$ elsewhere, respectively. The
_tree product_ is defined by

$\vcentcolon=\mathscr{J}(\tau_{1}\cdot\tau_{2}).$

######  Definition B.5 ([13, Section 4.3]).

For a decorated tree
$\tau=(T,\hat{T})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}$
and $k\in\mathbb{N}_{0}^{d}$, we write $\mathscr{I}_{k}(\tau)$ for a decorated
tree
$\sigma=(S,\hat{S})^{\tilde{\scaleobj{0.8}{\mathrm{N}}},\tilde{\mathfrak{o}}}_{\tilde{\mathfrak{e}}}$
obtained by connecting the old root $\rho_{\tau}$ to a new root
$\rho_{\sigma}$ with a new edge $e=(\rho_{0},\rho_{\tau})$ and by defining
$\hat{S}$, $\tilde{\scaleobj{0.8}{\mathrm{N}}}$, $\tilde{\mathfrak{o}}$ and
$\tilde{\mathfrak{e}}$ as an extension of $\hat{T}$,
$\scaleobj{0.8}{\mathrm{N}}$, $\mathfrak{o}$ and $\mathfrak{e}$ such that

$\hat{S}(\rho_{\sigma})=\tilde{\scaleobj{0.8}{\mathrm{N}}}(\rho_{\sigma})=\tilde{\mathfrak{o}}(\rho_{\sigma})=0,\quad\mathfrak{t}(e)=\mathscr{I},\quad\hat{S}(e)=0,\quad\tilde{\mathfrak{e}}(e)=k,$

For $i\in\\{1,\ldots,d\\}$, we write
$\mathscr{I}_{i}(\tau)\vcentcolon=\mathscr{I}_{\bm{e_{i}}}(\tau)$, where
$\bm{e}_{i}$ is the $i$th standard basis of $\mathbb{R}^{d}$.

######  Definition B.6 ([13, Definition 5.3]).

Let $\delta^{\prime}$ be the constant appearing in Assumption 3.10 and we fix
a small $\kappa\in(0,\delta^{\prime})$ such that $\delta+\kappa$ is
irrational. We assign the _degree_ to the types by222In Section 3, we set
$\lvert\Xi\rvert\vcentcolon=-2+\delta$, but the new definition (72) is more
convenient in Section C.6.

$\lvert\Xi\rvert\vcentcolon=-2+\delta+\kappa,\quad\lvert\mathscr{I}\rvert\vcentcolon=2.$
(72)

We extend the degree to $(k,v)\in\mathbb{Z}^{d}\oplus\mathbb{Z}[\mathfrak{L}]$
by

$\lvert(k,v)\rvert\vcentcolon=\sum_{i=1}^{d}k_{i}+a\lvert\Xi\rvert+b\lvert\mathscr{I}\rvert$

where $v=a\Xi+b\mathscr{I}$. For a decorated tree
$\tau=(F,\hat{F},\scaleobj{0.8}{\mathrm{N}},\mathfrak{o},\mathfrak{e})$, we
set

$\hat{E}_{i}\vcentcolon=\hat{F}^{-1}(i)\cap
E_{F},\quad\hat{E}\vcentcolon=\hat{E}_{1}\cup\hat{E}_{2},\quad\hat{N}_{i}\vcentcolon=\hat{F}^{-1}(i)\cap
N_{F}$

and we define two notions of degrees $\lvert\cdot\rvert_{-}$ and
$\lvert\cdot\rvert_{+}$ by

$\displaystyle\lvert\tau\rvert_{-}$ $\displaystyle\vcentcolon=\sum_{e\in
E_{F}\setminus\hat{E}}(\lvert\mathfrak{t}(e)\rvert-\mathfrak{e}(e))+\sum_{x\in
N_{F}}\scaleobj{0.8}{\mathrm{N}}(x)$ $\displaystyle\lvert\tau\rvert_{+}$
$\displaystyle\vcentcolon=\sum_{e\in
E_{F}\setminus\hat{E}_{2}}(\lvert\mathfrak{t}(e)\rvert-\mathfrak{e}(e))+\sum_{x\in
N_{F}}\scaleobj{0.8}{\mathrm{N}}(x)+\sum_{x\in
N_{F}\setminus\hat{N}_{2}}\lvert\mathfrak{o}(x)\rvert.$

######  Remark B.7.

Since Schauder’s estimate does not hold for integer exponents, we assume that
$\delta+\kappa$ is irrational.

###  B.2 Hopf algebras on forests and trees

In this section, we introduce Hopf algebra structures on some spaces of
forests and those of trees. For this purpose, we begin with introducing
contraction operators.

######  Definition B.8 ([13, Definition 3.18]).

We set

$(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}\vcentcolon=(\mathscr{K}_{\hat{F}}F,\hat{F})^{[\scaleobj{0.8}{\mathrm{N}}],[\mathfrak{o}]}_{[\mathfrak{e}]},$

where

  * •

$\mathscr{K}_{\hat{F}}F$ is the quotient forest $F/\sim$, where the equivalent
relation $\sim$ is in the sense of Definition B.3-(c);

  * •

$\hat{F}$ and $[\mathfrak{e}]$ are natural “restrictions”;

  * •

one has $[\scaleobj{0.8}{\mathrm{N}}](x)\vcentcolon=\sum_{y\sim
x}\scaleobj{0.8}{\mathrm{N}}(y)$;

  * •

one has

$[\mathfrak{o}](x)\vcentcolon=\sum_{y\sim x}\mathfrak{o}(y)+\sum_{e\in
E(x)}\mathfrak{t}(e),\quad E(x)\vcentcolon=\\{(y,z)\in
E\nonscript\>|\nonscript\>\mathopen{}y\sim z\sim x\\}.$

For a decorated forest $\tau$ and $i\in\mathbb{N}$, one has a unique
decomposition $\tau=\mu\cdot\nu$ such that on $\nu$ the map $\hat{F}$ is equal
to $i$ and on each component of $\mu$ the map $\hat{F}$ is not equal to $i$
everywhere.

######  Definition B.9.

Then, we set

$k_{i}(\nu)\vcentcolon=\begin{cases}(\bullet,i,\sum_{x\in
N_{\nu}}\scaleobj{0.8}{\mathrm{N}}(x),0,0)&\text{if }\sum_{x\in
N_{\nu}}\scaleobj{0.8}{\mathrm{N}}(x)>0\\\
\varnothing&\text{otherwise}\end{cases}$

and

$\mathscr{K}_{i}(\tau)\vcentcolon=\mathscr{K}(\mu)\cdot k_{i}(\nu).$

In addition, we denote by $\hat{\mathscr{K}}_{i}(\tau)$ the decorated forest
that is obtained from $\mathscr{K}_{i}(\tau)$ by setting $\mathfrak{o}$ to $0$
on $\hat{F}^{-1}(i)$.

######  Remark B.10.

$\nu$ is allowed to be an empty forest $\varnothing$ and
$k_{i}(\varnothing)=\varnothing$.

With these operators, one can write Hopf algebras associated to regularity
structures and renormalization structures.

######  Definition B.11.

We define vector spaces $H_{1},H_{\circ}$ as follows.

  1. (a)

We denote by the free vector space generated by

$(H_{1})\vcentcolon=\\{(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}\nonscript\>|\nonscript\>\mathopen{}\hat{F}\leq
1,\,\,\mathscr{K}_{1}(F)=F\\}.$

  2. (b)

We denote by the free vector space generated by $\mathfrak{B}(H_{\circ})$,
where $\tau\in\mathfrak{B}(H_{\circ})$ if and only if

     * •

$\tau$ is a tree and $\hat{F}\leq 1$;

     * •

$\mathscr{K}(\tau)=\tau$.

######  Definition B.12 ([13, Definition 3.3]).

Given a decorated forest
$\tau=(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}$,
we denote by $\mathfrak{U}_{1}(\tau)$ the set of all subforests of $F$ which
contains $\hat{F}^{-1}(1)$ and subforests of $F$ that are disjoint from
$\hat{F}^{-1}(2)$. We set

$\begin{multlined}\Delta_{1}\tau\vcentcolon=\sum_{A\in\mathfrak{U}_{1}(\tau)}\sum_{\scaleobj{0.8}{\mathrm{N}}_{A}:\scaleobj{0.8}{\mathrm{N}}_{A}\leq\scaleobj{0.8}{\mathrm{N}}}\sum_{\varepsilon^{F}_{A}}\frac{1}{\varepsilon^{F}_{A}!}\binom{\scaleobj{0.8}{\mathrm{N}}}{\scaleobj{0.8}{\mathrm{N}}_{A}}(A,F|_{A},\scaleobj{0.8}{\mathrm{N}}_{A}+\pi\varepsilon^{F}_{A},\mathfrak{o}|_{N_{A}},\mathfrak{e}|_{E_{A}})\\\
\otimes(F,\hat{F}\cup_{1}A,\scaleobj{0.8}{\mathrm{N}}-\scaleobj{0.8}{\mathrm{N}}_{A},\mathfrak{o}+\scaleobj{0.8}{\mathrm{N}}_{A}+\pi(\varepsilon^{F}_{A}-\mathfrak{e}\mathbbm{1}_{A}),\mathfrak{e}\mathbbm{1}_{E_{F}\setminus
E_{A}}+\varepsilon^{F}_{A}),\end{multlined}\Delta_{1}\tau\vcentcolon=\sum_{A\in\mathfrak{U}_{1}(\tau)}\sum_{\scaleobj{0.8}{\mathrm{N}}_{A}:\scaleobj{0.8}{\mathrm{N}}_{A}\leq\scaleobj{0.8}{\mathrm{N}}}\sum_{\varepsilon^{F}_{A}}\frac{1}{\varepsilon^{F}_{A}!}\binom{\scaleobj{0.8}{\mathrm{N}}}{\scaleobj{0.8}{\mathrm{N}}_{A}}(A,F|_{A},\scaleobj{0.8}{\mathrm{N}}_{A}+\pi\varepsilon^{F}_{A},\mathfrak{o}|_{N_{A}},\mathfrak{e}|_{E_{A}})\\\
\otimes(F,\hat{F}\cup_{1}A,\scaleobj{0.8}{\mathrm{N}}-\scaleobj{0.8}{\mathrm{N}}_{A},\mathfrak{o}+\scaleobj{0.8}{\mathrm{N}}_{A}+\pi(\varepsilon^{F}_{A}-\mathfrak{e}\mathbbm{1}_{A}),\mathfrak{e}\mathbbm{1}_{E_{F}\setminus
E_{A}}+\varepsilon^{F}_{A}),$ (73)

where

  * •

$\varepsilon^{F}_{A}$ runs over all maps $E_{F}\to\mathbb{N}_{0}^{d}$
supported on the (outgoing) boundary

$\partial(A,F)\vcentcolon=\\{(e_{+},e_{-})\in E_{F}\setminus
E_{A}\nonscript\>|\nonscript\>\mathopen{}e_{+}\in N_{A}\\};$

  * •

for $\varepsilon:E_{F}\to\mathbb{N}_{0}^{d}$ one defines
$\pi\varepsilon:N_{F}\to\mathbb{Z}^{d}$ by

$\pi\varepsilon(x)\vcentcolon=\sum_{\begin{subarray}{c}e\in E_{F}\\\
e=(x,y)\text{ for some }y\end{subarray}}\varepsilon(x);$

  * •

$\hat{F}\cup_{1}A$ is the map defined by

$\hat{F}\cup_{1}A(x)\vcentcolon=\begin{cases}1&\text{if }x\in A\\\
\hat{F}(x)&\text{otherwise}.\end{cases}$

Some of the main results from [13] are the following.

######  Proposition B.13 ([13, Proposition 4.11]).

The vector space $H_{1}$ is a Hopf algebra with multiplication

$\mathcal{M}(\tau_{1}\otimes\tau_{2})\vcentcolon=\mathscr{K}_{1}(\tau_{1}\cdot\tau_{2}),$

with unit $\varnothing$, with coproduct
$(\mathscr{K}_{1}\otimes\mathscr{K}_{1})\Delta_{1}$ and with counit

$\mathbbm{1}^{\prime}_{H_{1}}((F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}})\vcentcolon=\mathbbm{1}_{\\{\varnothing\\}}((F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}),\quad$

The Hopf algebra $H_{1}$ is graded with respect to $\lvert\cdot\rvert_{-}$.

######  Proposition B.14 ([13, Proposition 4.14]).

The vector space $H_{\circ}$ is a left comodule over the Hopf algebra $H_{1}$
with coaction

$(\mathscr{K}_{1}\otimes\mathscr{K})\Delta_{1}:H_{\circ}\to H_{1}\otimes
H_{\circ}.$

###  B.3 Rule

We set

$\vcentcolon=\\{(F,\hat{F},\scaleobj{0.8}{\mathrm{N}},\mathfrak{o},\mathfrak{e})\in\mathfrak{F}\nonscript\>|\nonscript\>\mathopen{}F\mbox{
is a tree, }\hat{F}\equiv 0,\,\,\mathfrak{o}\equiv 0\\}.$

The set $\mathfrak{T}$ is a monoid with the tree product and with the trivial
tree as unit. We simply write $T^{\scaleobj{0.8}{\mathrm{N}}}_{\mathfrak{e}}$
for $(T,0,\scaleobj{0.8}{\mathrm{N}},0,\mathfrak{e})\in\mathfrak{T}$.

######  Definition B.15.

Given a decorated tree
$T^{\scaleobj{0.8}{\mathrm{N}}}_{\mathfrak{e}}\in\mathfrak{T}$, we associate
to each $x\in N_{T}$ a node type

$\vcentcolon=\mathscr{N}(x)\vcentcolon=\big{(}(\mathfrak{t}(e_{1}),\mathfrak{e}(e_{1})),\ldots,(\mathfrak{t}(e_{n}),\mathfrak{e}(e_{n}))\big{)},$

where $(e_{1},\ldots,e_{n})$ are the edges leaving the node $x$, namely, for
each $j$ one can find a $y_{j}\in N_{T}$ such that $e_{j}=(x,y_{j})$.

######  Definition B.16.

Let $(\mathfrak{L}\times\mathbb{N}_{0})^{n}/\sim_{n}$ be the set of unordered
$n$-tuples valued in $\mathfrak{L}\times\mathbb{N}_{0}$ and let
$\mathcal{P}\mathcal{N}$ be the power set of
$\cup_{n\in\mathbb{N}_{0}}(\mathfrak{L}\times\mathbb{N}_{0})^{n}/\sim_{n}$. We
define the _rule_ $:\mathfrak{L}\to\mathcal{P}\mathcal{N}$ by

$\displaystyle R(\Xi)$ $\displaystyle\vcentcolon=\\{()\\},$ $\displaystyle
R(\mathscr{I})$
$\displaystyle\vcentcolon=\\{([\mathscr{I}]_{n}),([\mathscr{I}]_{n},\mathscr{I}_{i}),([\mathscr{I}]_{n},\mathscr{I}_{i},\mathscr{I}_{j}),([\mathscr{I}]_{n},\Xi);n\in\mathbb{N}_{0},i,j\in\\{1,\ldots,d\\}\\},$

where we write $[\mathscr{I}]_{n}$ for the $n$-tuple of $(\mathscr{I},0)$ and
write $\mathscr{I}_{i}$ for $(\mathscr{I},\bm{e}_{i})$, where $\bm{e}_{i}$ is
the $i$th unit vector in $\mathbb{R}^{d}$.

It is not difficult to show that the rule $R$ is subcritical in the sense of
[13, Definition 5.14] and complete in the sense of [13, Definition 5.20].

######  Definition B.17 ([13, Definition 5.8]).

Let $\tau=T^{\scaleobj{0.8}{\mathrm{N}}}_{\mathfrak{e}}\in\mathfrak{T}$.

  1. (a)

We say $\tau$ _conforms to the rule $R$ at the node $x$_ if the following
hold:

     * •

if $x$ is the root, then $\mathscr{N}(x)\in R(\Xi)$ or $\mathscr{N}(x)\in
R(\mathscr{I})$;

     * •

otherwise, one has $\mathscr{N}(x)\in R(\mathfrak{t}(e))$, where $e$ is the
edge such that $e=(y,x)$ for some node $y$.

  2. (b)

We say $\tau$ _conforms to the rule $R$_ if $\tau$ conforms to $R$ at every
node, except possibly the root.

  3. (c)

We say $\tau$ _strongly conforms to the rule $R$_ if $\tau$ conforms to $R$ at
every node.

######  Definition B.18 ([13, Definition 5.13]).

We define sets $\mathfrak{T}_{\diamond}$ ($\diamond\in\\{\circ,1,-\\}$) as
follows.

  1. (a)

We denote by $\subseteq\mathfrak{T}$ the set of trees which strongly conform
to $R$.

  2. (b)

We denote by $\subseteq\mathfrak{F}$ the smallest submonoid under the forest
product which contains $\mathfrak{T}_{\circ}$.

  3. (c)

We denote by $\subseteq\mathfrak{T}_{\circ}$ the set of trees
$T^{\scaleobj{0.8}{\mathrm{N}}}_{\mathfrak{e}}$ with the following properties:

     * •

one has $\lvert\tau\rvert_{-}<0$ and $\scaleobj{0.8}{\mathrm{N}}(\rho_{T})=0$;

     * •

if there exists only one edge containing $\rho_{T}$, then

$T^{\scaleobj{0.8}{\mathrm{N}}}_{\mathfrak{e}}=\leavevmode\hbox
to19.87pt{\vbox to48.83pt{\pgfpicture\makeatletter\hbox{\hskip
7.94322pt\lower-16.94461pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ } {{}}
{{{}{}{{}}{}}}{{{}}}{{{{}}{{}}}}{{}}{{{
}}}{{{{}}}}{}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{2.82861pt}{0.0pt}\pgfsys@curveto{2.82861pt}{1.56221pt}{1.56221pt}{2.82861pt}{0.0pt}{2.82861pt}\pgfsys@curveto{-1.56221pt}{2.82861pt}{-2.82861pt}{1.56221pt}{-2.82861pt}{0.0pt}\pgfsys@curveto{-2.82861pt}{-1.56221pt}{-1.56221pt}{-2.82861pt}{0.0pt}{-2.82861pt}\pgfsys@curveto{1.56221pt}{-2.82861pt}{2.82861pt}{-1.56221pt}{2.82861pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.61021pt}{-11.66718pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho_{T}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{2.82863pt}{28.45276pt}\pgfsys@curveto{2.82863pt}{30.01497pt}{1.56223pt}{31.28137pt}{0.00002pt}{31.28137pt}\pgfsys@curveto{-1.5622pt}{31.28137pt}{-2.8286pt}{30.01497pt}{-2.8286pt}{28.45276pt}\pgfsys@curveto{-2.8286pt}{26.89055pt}{-1.5622pt}{25.62415pt}{0.00002pt}{25.62415pt}\pgfsys@curveto{1.56223pt}{25.62415pt}{2.82863pt}{26.89055pt}{2.82863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.00002pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.00002pt}{28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{3.42862pt}\pgfsys@lineto{0.00002pt}{25.02414pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.93301pt}{12.07361pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$e$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\scaleobj{0.8}{\mathrm{N}}(\rho_{T})=\mathfrak{e}(e)=0,\,\,\mathfrak{t}(e)=\Xi.$
(74)

###  B.4 Definition of the regularity structure

The content of this section is parallel to [13, Section 5.5]. Our goal here is
to construct subspaces of $H_{\diamond}$ ($\diamond\in\\{1,\circ\\}$) which
provide a correct framework for the theory of regularity structures. Since we
desire that elements of those spaces conform to the rule $R$, one might want
to consider a subspace spanned by $\mathfrak{T}_{\diamond}$. However,
$\mathfrak{T}_{\diamond}$ is not closed under the coproduct of $H_{\diamond}$.
Therefore, we introduce the following definition.

######  Definition B.19.

Recall the notation introduced in Definition B.12. For
$\diamond\in\\{1,\circ\\}$, we denote by
$\mathfrak{B}(H^{C}_{\diamond})\subseteq\mathfrak{B}(H_{\diamond})$ the set
consisting of

$\mathscr{K}_{1}(F,\hat{F}\cup_{1}A,\scaleobj{0.8}{\mathrm{N}}-\scaleobj{0.8}{\mathrm{N}}_{A},\scaleobj{0.8}{\mathrm{N}}_{A}+\pi(\varepsilon^{F}_{A}-\mathfrak{e}\mathbbm{1}_{A}),\mathfrak{e}\mathbbm{1}_{F\setminus
A}+\varepsilon^{F}_{A})$ (75)

for
$\tau=(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},0}_{\mathfrak{e}}\in\mathfrak{T}_{\diamond}$,
$A\in\mathfrak{U}_{1}(\tau)$,
$\scaleobj{0.8}{\mathrm{N}}_{A}\leq\scaleobj{0.8}{\mathrm{N}}$ with
$\operatorname{supp}(\scaleobj{0.8}{\mathrm{N}}_{A})\subseteq N_{A}$ and
$\varepsilon^{F}_{A}:E_{F}\to\mathbb{N}_{0}^{d}$ with
$\operatorname{supp}(\varepsilon^{F}_{A})\subseteq\partial(A,F)$. We denote by
the free vector space generated by $\mathfrak{B}(H^{C}_{\diamond})$.

######  Remark B.20.

By choosing $A=\varnothing$ one observes
$\mathfrak{T}_{\diamond}\subseteq\mathfrak{B}(H^{C}_{\diamond})$ for
$\diamond\in\\{1,\circ\\}$. In fact, as Lemma B.21 below shows, $H^{C}_{1}$ is
the smallest subbialgebra of $H_{\diamond}$ both containing
$\mathfrak{T}_{\diamond}$ and closed under the coactions.

######  Lemma B.21.

The subspace $H_{1}^{C}$ is a subbialgebra of $H_{1}$. Furthermore, the
statements of Proposition B.14 remain valid if one replaces
$(H_{1},H_{\circ})$ by $(H^{C}_{1},H^{C}_{\circ})$.

###### Proof.

This is essentially proven in [13, Lemma 5.25 and Lemma 5.28]. ∎

######  Definition B.22.

For $\diamond\in\\{1,\circ\\}$ we denote by the free vector space generated by

$\mathfrak{B}(H^{R}_{\diamond})\vcentcolon=\\{(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}\in\mathfrak{B}(H^{C}_{\diamond})\nonscript\>|\nonscript\>\mathopen{}(F,\hat{F}\mathbbm{1}_{\hat{F}\neq
1})^{\scaleobj{0.8}{\mathrm{N}},0}_{\mathfrak{e}}\in\mathfrak{T}_{\diamond}\\}.$

######  Definition B.23.

We denote by $\mathscr{T}_{1}$ the free vector space generated by

$\mathfrak{B}(\mathscr{T}_{1})\vcentcolon=\\{\tau\in\mathfrak{B}(H^{C}_{1})\nonscript\>|\nonscript\>\mathopen{}(F,0)^{\scaleobj{0.8}{\mathrm{N}},0}_{\mathfrak{e}}\in\mathfrak{T}_{-}\text{
for every connected component
}(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}\text{ of
}\tau\\}$

and by $\mathscr{T}_{2}$ the free vector space generated by
$\mathfrak{B}(\mathscr{T}_{2})$, where $\tau\in\mathfrak{B}(\mathscr{T}_{2})$
if and only if

$\hat{\mathscr{K}}_{2}\big{[}\mathscr{I}_{k_{1}}(\tau_{1})\cdots\mathscr{I}_{k_{n}}(\tau_{n}){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}^{m}\big{]}$

for $n\in\mathbb{N}_{0}$, $k_{1},\ldots,k_{n},m\in\mathbb{N}_{0}^{d}$ and
$\tau_{1},\ldots,\tau_{n}\in H^{R}_{\circ}$ such that
$\lvert\mathscr{I}_{k_{j}}(\tau_{j})\rvert_{+}>0$ for every $j=1,\ldots,n$. We
denote by $\mathfrak{p}_{1}^{C}:H_{1}^{C}\to\mathscr{T}_{1}$ the natural
projection. We note that $\mathscr{T}_{1}$ is an algebra under the forest
product and $\mathscr{T}_{2}$ is an algebra under the tree product.

######  Proposition B.24 ([13, Proposition 5.35]).

The linear map

$\Delta_{1}\vcentcolon=(\mathfrak{p}_{1}^{C}\otimes\mathfrak{p}_{1}^{C})(\mathscr{K}_{1}\otimes\mathscr{K}_{1})\Delta_{1}:\mathscr{T}_{1}\to\mathscr{T}_{1}\otimes\mathscr{T}_{1}$

defines a coproduct over the algebra $\mathscr{T}_{1}$ (with the forest
product as multiplication). With this coproduct and the counit as in
Proposition B.13, $\mathscr{T}_{1}$ is a Hopf algebra. Furthermore, the vector
space $H^{R}_{\circ}$ is a right comodule over $\mathscr{T}_{1}$ with coaction

$\vcentcolon=(\mathfrak{p}_{1}^{C}\otimes\operatorname{Id})(\mathscr{K}_{1}\otimes\mathscr{K})\Delta_{1}:H^{R}_{\circ}\to\mathscr{T}_{1}\otimes
H^{R}_{\circ}.$

######  Remark B.25.

As shown in [13, Proposition 5.34], one can view $\mathscr{T}_{2}$ as a Hopf
algebra with grade $\lvert\cdot\rvert_{+}$ and one can define a coaction
$\Delta^{\circ}_{+}:H^{R}_{\circ}\to H^{R}_{\circ}\otimes\mathscr{T}_{2}$, of
which we do not need the precise definition here but will only use the
recursive formula [13, Proposition 4.17].

######  Definition B.26.

We set

$\mathscr{T}\vcentcolon=H^{R}_{\circ},\quad\mathscr{T}_{+}\vcentcolon=\mathscr{T}_{2},\quad\mathscr{T}_{-}\vcentcolon=\mathscr{T}_{1},$

Then, in the language of [6], the pair is a concrete regularity structure and
the pair is a renormalization structure for the generalized PAM. For
$\diamond\in\\{-,+\\}$, we denote

  * •

by the coproduct of $\mathscr{T}_{\diamond}$,

  * •

by the unit of $\mathscr{T}_{\diamond}$,

  * •

by the counit of $\mathscr{T}_{\diamond}$ and

  * •

by the antipode of $\mathscr{T}_{\diamond}$.

Recall that the product $\mathscr{M}_{-}$ of $\mathscr{T}_{-}$ is the forest
product while the product $\mathscr{M}_{+}$ of $\mathscr{T}_{+}$ is the tree
product. We write $\bm{1}\in\mathscr{T}$ for $\bullet^{0}$. For
$\diamond\in\\{-,+\\}$, the Hopf algebra $\mathscr{T}_{\diamond}$ is graded
with $\lvert\cdot\rvert_{\diamond}$. The vector space $\mathscr{T}$ is graded
both with $\lvert\cdot\rvert_{-}$ and with $\lvert\cdot\rvert_{+}$.

######  Definition B.27.

As shown in [13, Proposition 5.39], if

$A\vcentcolon=\\{\lvert\tau\rvert_{+}\nonscript\>|\nonscript\>\mathopen{}\tau\in\mathfrak{B}(\mathscr{T})\\},$

and we denote by $G$ the character group of $\mathscr{T}_{+}$, the triplet
$(A,\mathscr{T},G)$ is a regularity structure in the sense of [34, Definition
2.1]. We have the graded decomposition

$\mathscr{T}=\oplus_{\gamma\in
A}\mathscr{T}_{\gamma},\quad\mathscr{T}_{\gamma}\vcentcolon=\operatorname{span}\\{\tau\in\mathfrak{B}(\mathscr{T})\nonscript\>|\nonscript\>\mathopen{}\lvert\tau\rvert_{+}=\gamma\\}.$

We write for the natural projection from $\mathscr{T}$ to
$\mathscr{T}_{<\beta}\vcentcolon=\oplus_{\gamma<\beta}\mathscr{T}_{\gamma}$.

######  Definition B.28.

If $\tau,\sigma\in\mathfrak{B}(\mathscr{T})$ are such that
$\tau\sigma\in\mathfrak{B}(\mathscr{T})$, we write
$\tau\star\sigma\vcentcolon=\tau\sigma$. We extend the product $\star$
bilineary. Note that the product $\star$ is not defined for all pairs
$(\tau,\sigma)$.

The following lemma essentially states that the product $\star$ is regular in
the sense of [34, Definition 4.6].

######  Lemma B.29 ([13, Proposition 3.11]).

If $\tau,\sigma\in\mathfrak{B}(\mathscr{T})$ are such that
$\tau\sigma\in\mathfrak{B}(\mathscr{T})$, then
$\Delta^{\circ}_{+}(\tau\star\sigma)=\Delta^{\circ}_{+}(\tau)\Delta^{\circ}_{+}(\sigma)$.

######  Definition B.30.

Let $V$ be the subspace of $\mathscr{T}$ generated by

$\mathscr{I}(\tau_{1})\cdots\mathscr{I}(\tau_{n}),\qquad\tau_{1},\ldots,\tau_{n}\in\mathscr{T}.$

For $i\in\\{1,\ldots,d\\}$, we define the linear map $:V\to\mathscr{T}$,
called a _derivative_ , by

$\mathscr{D}_{i}[\mathscr{I}(\tau_{1})\cdots\mathscr{I}(\tau_{n})]=\sum_{j=1}^{d}\mathscr{I}_{i}(\tau_{j})\prod_{k\neq
j}\mathscr{I}(\tau_{k}).$

######  Proposition B.31 ([13, Section 6.1]).

Let $\mathfrak{i}_{-}:\mathscr{T}_{-}\to H^{R}_{1}$ be the natural projection.
Then, there exists a unique algebra morphism
$\hat{\mathscr{A}}_{-}:\mathscr{T}_{-}\to H^{R}_{1}$ such that

$\mathscr{M}_{H_{1}}(\hat{\mathscr{A}}_{-}\otimes\operatorname{Id}_{\hat{\mathscr{T}}_{-}})(\mathscr{K}_{1}\otimes\mathscr{K}_{1})\Delta_{1}\mathfrak{i}_{-}=\bm{1}^{\prime}_{-}(\cdot)\bm{1}_{H_{1}}\quad\text{on
}\mathscr{T}_{-},$

where $\mathscr{M}_{H_{1}}$ is the product in $H_{1}$.

######  Definition B.32.

We call a _negative twisted antipode_.

As for $\hat{\mathscr{A}}_{-}$, we only use the following property. As shown
in [13, Proposition 6.6], one has the following recursive formula:

$\hat{\mathscr{A}}_{-}\tau=-\mathscr{M}_{H_{1}}(\hat{\mathscr{A}}_{-}\otimes\operatorname{Id}_{H^{R}_{1}})(\hat{\Delta}_{-}\tau-\tau\otimes\bm{1}_{-}),$
(76)

where
$\hat{\Delta}_{-}\vcentcolon=(\mathfrak{p}_{1}^{C}\otimes\operatorname{Id}_{H^{R}_{1}})(\mathscr{K}_{1}\otimes\mathscr{K}_{1})\Delta_{1}$.

#### List of symbols from Appendix B

[type=symbols,style=index]

##  Appendix C Proof of Theorem 3.3

Based on the framework discussed in Appendix B, here we provide the details to
prove Theorem 3.3.

###  C.1 Modelled distributions

An important concept in the theory of regularity structures is the _modelled
distribution_ ([34, Definition 3.1]). We denote by
$\mathcal{D}^{\gamma}(\mathscr{T},\mathscr{Z})=\mathcal{D}^{\gamma}(\mathscr{Z})$
the space of modelled distributions with respect to the model $\mathscr{Z}$
whose images are in
$\mathscr{T}_{<\gamma}=\oplus_{\beta<\gamma}\mathscr{T}_{\beta}$. We set

$\mathcal{D}_{\alpha}^{\gamma}(\mathscr{T},\mathscr{Z})\vcentcolon=\\{f\in\mathcal{D}^{\gamma}(\mathscr{T},\mathscr{Z})\nonscript\>|\nonscript\>\mathopen{}f\mbox{
is }\oplus_{\alpha\leq\beta<\gamma}\mathscr{T}_{\beta}\mbox{-valued}\\}.$

We will use the norm
$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\cdot\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}$
given by [34, (3.1)].

######  Definition C.1.

Let $\mathscr{Z}$ be a model over $\mathscr{T}$ and let $\gamma,l>0$. By [34,
Theorem 3.10], there exists a unique continuous linear operator
$\mathcal{R}=\mathcal{R}^{\mathscr{Z}}:\mathcal{D}^{\gamma}(\mathscr{T},\mathscr{Z})\to\mathcal{C}^{\min
A}_{\operatorname{loc}}(\mathbb{R}^{d})$ with the following property: there
exists a $C=C(\gamma,l,\mathscr{T})>0$ such that for every compact set
$\mathfrak{K}\subseteq\mathbb{R}^{d}$

$\lvert(\mathcal{R}f-\Pi_{x}f(x))(\phi_{x}^{\lambda})\rvert\leq
C\lambda^{\gamma}\lVert\mathscr{Z}\rVert_{\gamma;B(\mathfrak{K},l)}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)},\quad\text{where
}\phi^{\lambda}_{x}\vcentcolon=\lambda^{-d}\phi(\lambda^{-1}(\cdot-x)),$ (77)

uniformly over $\phi\in\mathcal{C}^{2}(B(0,l))$ with
$\lVert\phi\rVert_{\mathcal{C}^{r}(\mathbb{R}^{d})}\leq 1$, $\lambda\in(0,1)$,
$f\in\mathcal{D}^{\gamma}(\mathscr{T},\mathscr{Z})$ and $x\in\mathfrak{K}$.
The operator $\mathcal{R}$ is called the _reconstruction operator_.

######  Proposition C.2 ([34, Theorem 4.7]).

Let $V_{1}$ and $V_{2}$ be subspaces of $\mathscr{T}$ closed under the action
of the structure group. Suppose the product $\tau_{1}\star\tau_{2}$ is well-
defined for every $\tau_{1}\in V_{1}$ and $\tau_{2}\in V_{2}$. Let
$\mathscr{Z}$ be a model for $\mathscr{T}$ and let
$f_{i}\in\mathcal{D}_{\alpha_{i}}^{\gamma_{i}}(V_{i},\mathscr{Z})$ for
$i=1,2$. Then, if we set
$\gamma\vcentcolon=\min\\{\gamma_{1}+\alpha_{2},\gamma_{2}+\alpha_{1}\\}$, one
has $\mathfrak{p}_{<\gamma}(f_{1}\star
f_{2})\in\mathcal{D}_{\alpha_{1}+\alpha_{2}}^{\gamma}(\mathscr{T},\mathscr{Z})$.
Moreover, there exists a constant $C\in(0,\infty)$ which depends only on
$\mathscr{T}$ such that

$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathfrak{p}_{<\gamma}(f_{1}\star
f_{2})\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}\leq
C(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma_{1}+\gamma_{2};\mathfrak{K}})^{2}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f_{1}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma_{1};\mathfrak{K}}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f_{2}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma_{2};\mathfrak{K}}\quad\text{for
every compact set }\mathfrak{K}\subseteq\mathbb{R}^{d}.$

######  Definition C.3.

Let $F\in C^{\infty}_{b}(\mathbb{R}^{d})$ and let $V$ be a subspace of
$\\{\tau\in\mathscr{T}\nonscript\>|\nonscript\>\mathopen{}\mathfrak{p}_{<0}\tau=0\\}$
that is closed under the product $\star$ and under the action of the structure
group. We define the map $F^{\star}:V\to V$ by

$F^{\star}(\tau)\vcentcolon=\sum_{k\in\mathbb{N}_{0}}\frac{D^{k}F(\bar{\tau})}{k!}(\tau-\bar{\tau})^{\star
k},\quad\bar{\tau}\vcentcolon=\mathfrak{p}_{0}\tau.$

According to [34, Theorem 4.16], if $\gamma>0$ and
$f\in\mathcal{D}^{\gamma}(V,\mathscr{Z})$, then one has

$F^{\star}_{\gamma}(f)(x)\vcentcolon=\mathfrak{p}_{<\gamma}F^{\star}(f(x))\in\mathcal{D}^{\gamma}(V,\mathscr{Z}).$

Furthermore, there exist a constant $C\in(0,\infty)$ and an integer
$k\in\mathbb{N}$, which depend only on $\mathscr{T}$, $F$ and $\gamma$, such
that

$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathfrak{p}_{<\gamma}F(f)\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}\leq
C(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma:\mathfrak{K}})^{k}\quad\text{for
every compact set }\mathfrak{K}\subseteq\mathbb{R}^{d}.$ (78)

###  C.2 Operations with kernels

In the rest, we fix an admissible kernel $K$ as in Section 3.1.2. Recall the
notation $G_{N}$ from Definition 3.1 and we set $H_{N}\vcentcolon=G_{N}-K$.

######  Lemma C.4.

For every $N\in\mathbb{N}_{0}$, the function $H_{N}$ belongs to
$\mathcal{S}(\mathbb{R}^{d})$.

###### Proof.

Since the Fourier transform of $G_{N}-G$ has a compact support, we observe
that $H_{N}=(G_{N}-G)+(G-K)$ is smooth. Thus, it comes down to showing that
$H_{N}$ decays rapidly or equivalently, as $K$ is supported on $B(0,1)$, to
showing that $G_{N}$ decays rapidly. For $m\in\mathbb{N}_{0}^{d}$, one has

$\partial^{m}G_{N}=\mathcal{F}^{-1}[(1-\check{\chi})(2^{-N}\cdot)\lvert\cdot\rvert^{-2}(2\pi\mathrm{i}\cdot)^{m}]$

for some polynomial $P_{m}$. Then, for
$n=(n_{1},\ldots,n_{d})\in\mathbb{N}_{0}^{d}$,

$x^{n}\partial^{m}G_{N}=(2\pi
i)^{n}\mathcal{F}^{-1}[(1-\check{\chi})(2^{-N}\cdot)\partial^{n}(\lvert\cdot\rvert^{-2}P_{m})]+R,$

where $R\in\mathcal{S}(\mathbb{R}^{d})$. Therefore, if $n_{1},\ldots,n_{d}$
are sufficiently large, $\partial^{n}(\lvert\cdot\rvert^{-2}P_{m})$ is
integrable. This means $x^{n}\partial^{m}G_{N}$ is bounded and hence $G_{N}$
decays rapidly. ∎

######  Remark C.5.

Thanks to Lemma C.4, the convolution $H_{N}\ast f$ is well-defined for
$f\in\mathcal{S}^{\prime}$ and the distribution $H_{N}\ast f$ represents a
smooth function.

######  Definition C.6.

For $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, $N\in\mathbb{N}_{0}$, we set

$[\Delta(G_{N}-G)]\ast