dom}(\mathcal{Q}_{-\mathfrak{L}})$,

$\|(-\mathfrak{L})^{-\frac{1}{2}}[g]\|_{L^{2}(\mathbb{R}^{N})}\leq
C\|g\|_{L^{(2^{*})^{\prime}}(\mathbb{R}^{N})}.$

######  Remark 6.10.

The above assumption on the Green function is stronger than (2). We refer the
reader to [88] for a discussion on the validity of such an upper bound.

###### Proof of Lemma 6.9.

This is essentially Theorem 7.5 in [30], which we restate here for
completeness.

A direct calculation gives

$\begin{split}&\|(-\mathfrak{L})^{-\frac{1}{2}}[f]\|_{L^{2}(\mathbb{R}^{N})}^{2}=\int_{\mathbb{R}^{N}}(-\mathfrak{L})^{-\frac{1}{2}}[f](-\mathfrak{L})^{-\frac{1}{2}}[f]\,\mathrm{d}x=\int_{\mathbb{R}^{N}}f(-\mathfrak{L})^{-1}[f]\,\mathrm{d}x\\\
&=\int_{\mathbb{R}^{N}}f(x)\bigg{(}\int_{\mathbb{R}^{N}}\mathbb{G}_{-\mathfrak{L}}^{x}(y)f(y)\,\mathrm{d}y\bigg{)}\,\mathrm{d}x\leq
C\int_{\mathbb{R}^{N}}f(x)\bigg{(}\int_{\mathbb{R}^{N}}|x-y|^{-(N-\alpha)}f(y)\,\mathrm{d}y\bigg{)}\,\mathrm{d}x\\\
&=C\|(-\Delta)^{-\frac{\alpha}{4}}[f]\|_{L^{2}(\mathbb{R}^{N})}^{2}.\end{split}$

The classical Hardy-Littlewood-Sobolev

$\|(-\Delta)^{-\frac{\alpha}{4}}[f]\|_{L^{2}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{(2^{*})^{\prime}}(\mathbb{R}^{N})}$

then provides the result. ∎

###  6.2. The nonlinear case ($m>1$)

While in the linear case, the Nash method works perfectly, in the nonlinear
case, the Nash method simply does not work since the “nonlinear heat
semigroup” is not symmetric. On the other hand, the Moser iteration, which
provides an alternative proof in the linear case, can be adapted to work also
in the nonlinear case, and it shows how to prove smoothing effects from GNS
inequalities also in the nonlinear setting. However GNS are not sufficient to
perform Moser iteration in the nonlocal setting, another ingredient is needed:
the so-called Stroock-Varopoulos inequalities. Let us briefly explain how this
works.

$\begin{array}[]{cc}\text{Hom. nonlinear}\\\
L^{1}\text{--}L^{\infty}\text{-smoothing}\end{array}$GNS inequalitySobolev
inequality$\begin{array}[]{ll}\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\\\
\lesssim|x-x_{0}|^{-(N-\alpha)}\end{array}$$\begin{array}[]{cc}\text{Off-
diagonal}\\\ \text{heat kernel
bounds}\end{array}$$\begin{array}[]{cc}\text{Hom. linear}\\\
L^{1}\text{--}L^{\infty}\text{-smoothing}\end{array}$$\begin{array}[]{cc}\text{On-
diagonal}\\\ \text{heat kernel
bounds}\end{array}$$\begin{array}[]{cc}\eqref{G_3}\\\
\mathbb{G}_{I-\mathfrak{L}}^{x_{0}}\in
L^{p}(\mathbb{R}^{N})\end{array}$$\begin{array}[]{cc}\text{Nonhom.
nonlinear}\\\
L^{1}\text{--}L^{\infty}\text{-smoothing}\end{array}$[7]$\begin{array}[]{cc}\text{S.-V.}\\\
\text{and Moser}\end{array}$[7]$\begin{array}[]{cc}L^{p}\text{-}\\\
\text{interp.}\end{array}$Via
HLSIntegrationDef.Nash[7]$\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(\cdot,0)=\delta_{x_{0}}$$\begin{array}[]{cc}\text{Representation}\\\
\text{formula}\end{array}$IntegrationTheorem
3.5$\begin{array}[]{cc}\text{Homogeneous}\\\ \text{operator}\end{array}$
Figure 5. Implications in the nonlinear case. Note that still the off-diagonal
heat kernel bounds provide the strongest piece of information. However, we
also see that because of (2), on-diagonal heat kernel bounds ensure a closed
loop in the nonlinear case, assuming that [7] applies.

Assume that there exists $2^{*}\geq 2$ such that the Sobolev-Poincaré type
inequality holds ($2^{*}=2$ being the Poincaré case)

$\|f\|^{2}_{L^{2^{*}}(\mathbb{R}^{N})}\leq
C\int_{\mathbb{R}^{N}}f(-\mathfrak{L})f\,\mathrm{d}x=C\mathcal{Q}_{-\mathfrak{L}}[f]=C\|(-\mathfrak{L})^{\frac{1}{2}}[u]\|^{2}_{L^{2}(\mathbb{R}^{N})},$

where the last equality is true whenever the operator $-\mathfrak{L}$ has an
extension to $L^{2}(\mathbb{R}^{N})$. By simple interpolation of
$L^{p}$-norms, for $\tilde{p}\in[(1+m)/m,2)$ and
$\tilde{q}\in[1/m,\tilde{p})$,

$\|f\|_{L^{\tilde{p}}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{N})}^{\vartheta}\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}(1-\vartheta)}\quad\text{where}\quad\vartheta:=\vartheta(2^{*}).$

When dealing with energy estimates in the local case, a calculus equality
allows us to do the Moser iteration:

$\int_{\mathbb{R}^{N}}u^{p-1}(-\Delta)[u^{m}]\,\mathrm{d}x=\int_{\mathbb{R}^{N}}\nabla
u^{p-1}\cdot\nabla
u^{m}\,\mathrm{d}x=\frac{4m(p-1)}{(p+m-1)^{2}}\int_{\mathbb{R}^{N}}\big{|}\nabla
u^{\frac{p+m-1}{2}}\big{|}^{2}\,\mathrm{d}x.$

However, we just need an inequality, which in the nonlocal case has been
proven by Stroock and Varopoulos [107, 111] (cf. [37, Proposition 4.11] or
[64, Lemma 4.10]): For the same constant as above,

$\int_{\mathbb{R}^{N}}u^{p-1}(-\mathfrak{L})[u^{m}]\,\mathrm{d}x\gtrsim\int_{\mathbb{R}^{N}}u^{\frac{p+m-1}{2}}(-\mathfrak{L})[u^{\frac{p+m-1}{2}}]\,\mathrm{d}x\eqsim\big{\|}(-\mathfrak{L})^{\frac{1}{2}}[u^{\frac{p+m-1}{2}}]\big{\|}^{2}_{L^{2}(\mathbb{R}^{N})}.$

Combining the two above inequalities, one gets

(M) $\int
u^{p-1}(-\mathfrak{L})[u^{m}]\geq\frac{4m(p-1)}{(p+m-1)^{2}}\mathcal{Q}_{-\mathfrak{L}}[u^{\frac{p+m-1}{2}}]\geq\frac{4m(p-1)}{C^{2}(p+m-1)^{2}}\frac{\|u\|_{L^{\tilde{p}\frac{p+m-1}{2}}}^{\frac{2}{1-\vartheta}\frac{p+m-1}{2}}}{\|u\|_{L^{\tilde{q}\frac{p+m-1}{2}}}^{\frac{2\vartheta}{1-\vartheta}\frac{p+m-1}{2}}}.$

The above condition is the key to prove the following:

######  Theorem 6.11 (Green functions satisfying (2)).

Assume ($\textup{H}_{m}$). Then the following statements are equivalent:

  1. (i)

($L^{1}$–$L^{\infty}$-smoothing) Let $u$ be a weak dual solution of (GPME)
with initial data $u_{0}\in L^{1}(\mathbb{R}^{N})$, then

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq
Ct^{-N\theta_{1}}\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}^{\alpha\theta_{1}},$

where $\theta_{1}=(\alpha+N(m-1))^{-1}$.

  2. (ii)

(Subcritical GNS) For $\tilde{p}\in[(1+m)/m,2)$ and
$\tilde{q}\in[1/m,\tilde{p})$, and for all $f\in
L^{\tilde{q}}(\mathbb{R}^{N})\cap{\rm dom}(\mathcal{Q}_{-\mathfrak{L}})$ we
have

$\|f\|_{L^{\tilde{p}}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{N})}^{\vartheta}\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}(1-\vartheta)}\quad\text{where}\quad\vartheta:=\frac{\tilde{q}}{\tilde{p}}\frac{2^{*}-\tilde{p}}{2^{*}-\tilde{q}}.$

When we have at our disposal Green functions satisfying (2) then, by Theorem
3.1(a), we have the above nonlinear smoothing effect. This turns out to be
equivalent to a family of subcritical Gagliardo-Nirenberg-Sobolev, which then
are equivalent to Sobolev, see Lemma 6.15 below. In order to show that
subcritical GNS imply nonlinear smoothing, we will perform a Moser iteration.
We refer to [28, Section 3] for a more detailed exposition of the Moser
iteration in the fast diffusion case $0<m<1$, in the context of bounded
domains. It also contains a detailed discussion about the Green function
method versus the Moser iteration.

When we have integrable Green functions (2), we can obtain absolute bounds
(i.e. independent of the initial datum), as in the case of bounded domains
[34, 35, 24]. Such bounds imply weak GNS inequalities, which are equivalent to
Poincaré inequalities (Lemma 6.15). We notice that it is not possible (to the
best of our knowledge) to prove the converse implication via the Moser
iteration. In fact, the constant simply blows up at the limit $p\to\infty$. A
similar discussion can be found in [75, 77]. However we have seen in Theorem
3.7 a simple proof of the absolute bounds with the Green function method, so
that we can conclude that integrable Green functions imply Poincaré-type
inequalities as follows:

######  Proposition 6.12 (Green functions satisfying (2)).

Assume ($\textup{H}_{m}$). Given the following statements:

  1. (i)

(Absolute bound) Let $u$ be a weak dual solution of (GPME) with initial data
$u_{0}$, then

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq Ct^{-1/(m-1)}.$

  2. (ii)

(Subcritical GNS) For $\tilde{p}\in[(1+m)/m,2)$ and
$\tilde{q}\in[1/m,\tilde{p})$, and for all $f\in
L^{\tilde{q}}(\mathbb{R}^{N})\cap{\rm dom}(\mathcal{Q}_{-\mathfrak{L}})$ we
have

$\|f\|_{L^{\tilde{p}}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{N})}^{\vartheta}\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}(1-\vartheta)}\quad\text{where}\quad\vartheta:=\frac{\tilde{q}}{\tilde{p}}\frac{2-\tilde{p}}{2-\tilde{q}}.$

Then (i)$\Longrightarrow$(ii).

In order to prove the above theorem and proposition, we need a few results
that we prefer to state and prove separately since they have their own
interest. We shall start with the fact that (any) GNS is equivalent to some
$L^{q}$–$L^{p}$-smoothing. This can be directly seen by the Stroock-Varopoulos
inequality.

######  Proposition 6.13 ($L^{q}$–$L^{p}$-smoothing VS subcritical GNS).

Assume ($\textup{H}_{m}$).

  1. (a)

(Green functions satisfying (2)) The following statements are equivalent:

    1. (i)

($L^{q}$–$L^{p}$-smoothing) For $p\in[1+m,\infty)$ and $q\in[1,p)$, let $u$ be
a weak dual solution of (GPME) with initial data $u_{0}\in
L^{q}(\mathbb{R}^{N})$, then

$\|u(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\leq
C\bigg{(}\frac{(p+m-1)^{2}}{4m(m-1)(p-1)}\frac{1}{t}\bigg{)}^{\frac{N(p-q)\theta_{q}}{p}}\|u_{0}\|_{L^{q}(\mathbb{R}^{N})}^{\frac{q}{p}\frac{\theta_{q}}{\theta_{p}}},$

where $\theta_{r}:=(\alpha r+N(m-1))^{-1}$ and $C>0$ is independent of $p,q$.

    2. (ii)

(Subcritical GNS) For $\tilde{p}\in[(1+m)/m,2)$ and
$\tilde{q}\in[1/m,\tilde{p})$, and for all $f\in
L^{\tilde{q}}(\mathbb{R}^{N})\cap{\rm dom}(\mathcal{Q}_{-\mathfrak{L}})$ we
have

$\|f\|_{L^{\tilde{p}}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{N})}^{\vartheta}\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}(1-\vartheta)}\quad\text{where}\quad\vartheta:=\frac{\tilde{q}}{\tilde{p}}\frac{2^{*}-\tilde{p}}{2^{*}-\tilde{q}}.$

  2. (b)

(Green functions satisfying (2)) The following statements are equivalent:

    1. (i)

($L^{q}$–$L^{p}$-smoothing) For $p\in[1+m,\infty)$ and $q\in[1,p)$, let $u$ be
a weak dual solution of (GPME) with initial data $u_{0}\in
L^{q}(\mathbb{R}^{N})$, then

$\|u(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\leq
C\bigg{(}\frac{(p+m-1)^{2}}{4m(m-1)(p-1)}\frac{1}{t}\bigg{)}^{\frac{p-q}{p(m-1)}}\|u_{0}\|_{L^{q}(\mathbb{R}^{N})}^{\frac{q}{p}},$

where $C>0$ is independent of $p,q$.

    2. (ii)

(Subcritical GNS) For $\tilde{p}\in[(1+m)/m,2)$ and
$\tilde{q}\in[1/m,\tilde{p})$, and for all $f\in
L^{\tilde{q}}(\mathbb{R}^{N})\cap{\rm dom}(\mathcal{Q}_{-\mathfrak{L}})$ we
have

$\|f\|_{L^{\tilde{p}}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{N})}^{\vartheta}\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}(1-\vartheta)}\quad\text{where}\quad\vartheta:=\frac{\tilde{q}}{\tilde{p}}\frac{2-\tilde{p}}{2-\tilde{q}}.$

######  Remark 6.14.

  1. (a)

Let us make some comments on part (a). First of all, note that $\vartheta$ is
nothing but the standard quantity appearing in interpolation between
$L^{p}$-norms with $\tilde{q}<\tilde{p}<2^{*}$. We also see that when formally
$m\to 1^{-}$, $\tilde{p}=2$ and $\tilde{q}=1$ in the above subcritical GNS
inequality, and we recover the critical Nash inequality (see Section 6.1). In
our case $m>1$, and that is why we call it subcritical. Note, however, that
the standard GNS inequality (see Remark 6.5) is not included as a special case
here.

  2. (b)

The proof reveals that GNS inequalities always imply $L^{q}$–$L^{p}$-smoothing
effects, but the opposite implication requires further assumptions. Actually,
the equivalence which is always true is the one between
$L^{1}$–$L^{m+1}$-smoothing effects and subcritical Nash inequalities. In
fact, even operators only yielding boundedness estimates in the form of
Theorem 3.5 (see also Theorem 5.1), still enjoy the latter equivalence.

To provide a proof, we need:

######  Lemma 6.15 ([7, 102]).

Assume ($\textup{H}_{m}$) and $f\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$.
Then:

  1. (a)

(Sobolev) The following statements are equivalent:

    1. (i)

(Sobolev)

$\|f\|_{L^{2^{*}}(\mathbb{R}^{N})}\leq
C\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}}.$

    2. (ii)

(Subcritical GNS) For $\tilde{p}\in[(1+m)/m,2)$ and
$\tilde{q}\in[1/m,\tilde{p})$,

$\|f\|_{L^{\tilde{p}}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{N})}^{\vartheta}\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}(1-\vartheta)}\quad\text{where}\quad\vartheta:=\frac{\tilde{q}}{\tilde{p}}\frac{2^{*}-\tilde{p}}{2^{*}-\tilde{q}}.$

  2. (b)

(Poincaré) The following statements are equivalent:

    1. (i)

(Poincaré)

$\|f\|_{L^{2}(\mathbb{R}^{N})}\leq
C\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}}.$

    2. (ii)

(Subcritical GNS) For $\tilde{p}\in[(1+m)/m,2)$ and
$\tilde{q}\in[1/m,\tilde{p})$,

$\|f\|_{L^{\tilde{p}}(\mathbb{R}^{N})}\leq
C\|f\|_{L^{\tilde{q}}(\mathbb{R}^{N})}^{\vartheta}\mathcal{Q}_{-\mathfrak{L}}[f]^{\frac{1}{2}(1-\vartheta)}\quad\text{where}\quad\vartheta:=\frac{\tilde{q}}{\tilde{p}}\frac{2-\tilde{p}}{2-\tilde{q}}.$

######  Remark 6.16.

  1. (a)

In both cases, we simply check that $q$ in [7] is respectively given by
$2^{*}$ and $2$. It is also worth noting, that “any” family of
Sobolev/Poincaré-type inequalities is equivalent with the Sobolev/Poincaré
inequality. As a consequence, subcritical GNS inequalities are equivalent with
subcritical Nash inequalities, and then also equivalent with the standard Nash
and GNS inequalities, respectively. However, the subcritical ones might be
easier to prove in the nonlinear setting.

  2. (b)

The case $q=2$ is kind of curious since it yields a Poincaré inequality in
$\mathbb{R}^{N}$. Such an inequality is not fulfilled in e.g. the case
$\mathfrak{L}=\Delta$ as the spectrum is nonnegative. Hence, it provides the
intuition that absolute bounds holds if the spectrum of the operator is
positive (e.g. as in the case $-\mathfrak{L}=(I-\Delta)^{\frac{\alpha}{2}}$).

###### Proof of Proposition 6.13.

(a) (i)$\Longrightarrow$(ii). By the particular choices $1=q<p=m+1$, we
immediately have the corresponding $L^{1}$–$L^{m+1}$-smoothing effect. Then,
direct and formal888We have decided to present the differential version of
these estimates because the main idea is easier to follow. This is rigorous
for strong solutions, for instance when $\partial_{t}u\in L^{1}$. It often
happens that bounded weak solutions are strong, possibly under some additional
assumptions on $\mathfrak{L}$, see [59]. These formal computations can be
justified rigorously in several different ways. One possibility is to use
Steklov averages and Grönwall-type inequalities. It is beyond the scope of
this paper to justify these computations, but we remark that they can be shown
to hold for the mild solutions constructed in Appendix E, through standard
approximations. The energy computations, namely the ones involving the
$L^{m+1}$ norm, are always true for weak energy solutions (of which mild
solutions are a subclass of). computations show that

(6.1) $\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int
u^{m+1}=\int\partial_{t}(u^{m+1})=(m+1)\int u^{m}\partial_{t}u=-(m+1)\int
u^{m}(-\mathfrak{L})[u^{m}],$

and also, since $\mathfrak{L}$ is symmetric and $u$ solves (GPME),

(6.2) $\begin{split}\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int
u^{m}(-\mathfrak{L})[u^{m}]&=2\int\partial_{t}(u^{m})(-\mathfrak{L})[u^{m}]=2m\int
u^{m-1}\partial_{t}u(-\mathfrak{L})[u^{m}]\\\ &=-2m\int
u^{m-1}\Big{(}(-\mathfrak{L})[u^{m}]\Big{)}^{2}\leq 0.\end{split}$

Hence, by (6.2), we obtain in (6.1) that

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int u^{m+1}\geq-(m+1)\int
u_{0}^{m}(-\mathfrak{L})[u_{0}^{m}]=-(m+1)\mathcal{Q}_{-\mathfrak{L}}[u_{0}^{m}].$

We then integrate over $(0,T)$ and use $L^{1}$–$L^{m+1}$-smoothing effect to
get

$\begin{split}-(m+1)Q_{-\mathfrak{L}}[u_{0}^{m}]T&\leq\|u(T)\|_{L^{m+1}}^{m+1}-\|u_{0}\|_{L^{m+1}}^{m+1}\leq
CT^{-N((m+1)-1)\theta_{1}}\|u_{0}\|_{L^{1}}^{\frac{\theta_{1}}{\theta_{m+1}}}-\|u_{0}\|_{L^{m+1}}^{m+1},\end{split}$

or, by taking $f:=u_{0}^{m}$,

$\|f\|_{L^{\frac{m+1}{m}}}^{\frac{m+1}{m}}\leq
F(T):=C\|f\|_{L^{\frac{1}{m}}}^{\frac{1}{m}\frac{\theta_{1}}{\theta_{m+1}}}T^{-N((m+1)-1)\theta_{1}}+(m+1)Q_{-\mathfrak{L}}[f]T.$

The inequality is still valid if we infimize $F$ over $T>0$, and this gives
the subcritical Nash inequality, i.e., $\tilde{p}=(1+m)/m$ and $\tilde{q}=1/m$
in the stated subcritical GNS. Since the subcritical Nash inequality is a
subfamily of the subcritical GNS, it is equivalent with the Sobolev inequality
and then equivalent with GNS by Lemma 6.15.

(a) (ii)$\Longrightarrow$(i). Note that (ii) with $f=u^{\frac{p+m-1}{2}}$ and
the Stroock-Varopoulos inequality gives (M). Direct and formal999Again, it is
beyond the scope of this paper to justify these computations, but they hold
for e.g. the mild solutions constructed in Appendix E under some possibly
additional assumptions on $\mathfrak{L}$. See also the previous footnote.
calculations and the $L^{p}$-decay (Proposition 4.1(b)(ii)) give

(6.3) $\begin{split}\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int
u^{p}&=\int\partial_{t}(u^{p})=p\int u^{p-1}\partial_{t}u=-p\int
u^{p-1}(-\mathfrak{L})[u^{m}]\\\
&\leq-\frac{4mp(p-1)}{C^{2}(p+m-1)^{2}}\frac{\|u\|_{L^{\tilde{p}\frac{p+m-1}{2}}}^{\frac{2}{1-\vartheta}\frac{p+m-1}{2}}}{\|u_{0}\|_{L^{\tilde{q}\frac{p+m-1}{2}}}^{\frac{2\vartheta}{1-\vartheta}\frac{p+m-1}{2}}}=-\frac{4mp(p-1)}{C^{2}(p+m-1)^{2}}\frac{\|u\|_{L^{p}}^{1+\sigma}}{\|u_{0}\|_{L^{q}}^{\frac{2(2^{*}-\tilde{p})}{2^{*}(\tilde{p}-\tilde{q})}q}}\end{split}$

Where we have chosen:

$\tilde{p}:=\frac{2p}{p+m-1},\qquad\tilde{q}:=\frac{2q}{p+m-1}\qquad\mbox{and}\qquad\sigma:=\frac{2^{*}(2-\tilde{p})+\tilde{q}(2^{*}-2)}{2^{*}(\tilde{p}-\tilde{q})},$

and this choice is consistent with our assumptions. We also have that

$\frac{2}{1-\vartheta}\frac{p+m-1}{2}=\frac{2(2^{*}-\tilde{q})}{2^{*}(\tilde{p}-\tilde{q})}p\qquad\text{and}\qquad\frac{2\vartheta}{1-\vartheta}\frac{p+m-1}{2}=\frac{2(2^{*}-\tilde{p})}{2^{*}(\tilde{p}-\tilde{q})}q.$

so that, integrating the differential inequality we get (i). The proof of part
(a) is concluded.

(b) (i)$\Longrightarrow$(ii). Follows by a similar argument as in (a)
(i)$\Longrightarrow$(ii), except that

$F(T):=C\|f\|_{L^{\frac{1}{m}}}^{\frac{1}{m}}T^{-\frac{m}{m-1}}+(m+1)Q_{-\mathfrak{L}}[f]T.$

(b) (ii)$\Longrightarrow$(i). We argue exactly as in (a)
(ii)$\Longrightarrow$(i), but now

$\frac{2}{1-\vartheta}\frac{p+m-1}{2}=\frac{2-\tilde{q}}{\tilde{p}-\tilde{q}}p,\qquad\frac{2\vartheta}{1-\vartheta}\frac{p+m-1}{2}=\frac{2-\tilde{p}}{\tilde{p}-\tilde{q}}q,$
$\frac{1}{\sigma}=\frac{p-q}{m-1},\qquad\text{and}\qquad\frac{2-\tilde{p}}{\tilde{p}-\tilde{q}}\frac{1}{\sigma}=1.$

This yields the desired estimate. ∎

###### Proof of Theorem 6.11.

(ii)$\Longrightarrow$(i). In what follows, we will just sketch the essential
parts of the proof, in order to focus on the main ideas. The proof can
moreover be made rigorous by standard approximation techniques. Let us first
remark that it is enough to prove the following
$L^{m+1}$–$L^{\infty}$-smoothing effect:

(6.4) $\|u(t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq
Ct^{-N\theta_{m+1}}\|u_{0}\|_{L^{m+1}(\mathbb{R}^{N})}^{\alpha(m+1)\theta_{m+1}}.$

Indeed, the claimed $L^{1}$–$L^{\infty}$-smoothing effects is then deduced by
applying Lemma B.2 of Appendix B.

In order to prove the smoothing effect (6.4), we will iterate “Moser style”
the $L^{p}$–$L^{q}$-smoothing effects Proposition 6.13(a)(i): Let us define
$p_{0}=m+1$ and $p_{k}=2^{k}p_{0}$ for each $k\geq 1$, and $t_{k}$ such that
$t_{k}-t_{k-1}=\frac{t-t_{0}}{2^{k}}$, so that inequality Proposition
6.13(a)(i) becomes

$\displaystyle\|u(t_{k})\|_{L^{p_{k}}}\leq
I_{k}^{\frac{N(p_{k}-p_{k-1})}{p_{k}}\theta_{k-1}}\|u(t_{k-1})\|_{L^{p_{k-1}}}^{\frac{p_{k-1}\,\theta_{k-1}}{p_{k}\,\theta_{k}}}\qquad\mbox{with}\qquad
I_{k}\eqsim\frac{p_{k}}{t_{k}-t_{k-1}}\eqsim 4^{k}$

where $\theta_{k}:=\theta_{p_{k}}=(\alpha p_{k}+N(m-1))^{-1}$. More precisely,
we have that we can estimate $I_{k}$ uniformly as follows:

$I_{k}:=C\frac{(p_{k}+m-1)^{2}}{4m(m-1)(p_{k}-1)}\frac{1}{t_{k}-t_{k-1}}\leq
4^{k}\frac{C}{t-t_{0}},$

for some constant $C>0$ that depends only on $m,N$.

Then we iterate

$\displaystyle\|u(t_{k})\|_{L^{p_{k}}}$ $\displaystyle\leq
I_{k}^{\frac{N(p_{k}-p_{k-1})}{p_{k}}\theta_{k-1}}\|u(t_{k-1})\|_{L^{p_{k-1}}}^{\frac{p_{k-1}\,\theta_{k-1}}{p_{k}\,\theta_{k}}}$
$\displaystyle\leq
I_{k}^{\frac{N(p_{k}-p_{k-1})}{p_{k}}\theta_{k-1}}I_{k-1}^{\frac{N(p_{k-1}-p_{k-2})}{\cancel{p_{k-1}}}\theta_{k-2}\frac{\cancel{p_{k-1}}\theta_{k-1}}{p_{k}\theta_{k}}}\|u(t_{k-2})\|_{L^{p_{k-2}}}^{\frac{p_{k-2}\theta_{k-2}}{\cancel{p_{k-1}}\cancel{\theta_{k-1}}}\frac{\cancel{p_{k-1}}\,\cancel{\theta_{k-1}}}{p_{k}\,\theta_{k}}}$
$\displaystyle\;\;\vdots$
$\displaystyle\leq\prod_{j=1}^{k}I_{j}^{\frac{N(p_{j}-p_{j-1})}{p_{k}}\frac{\theta_{j}\theta_{j-1}}{\theta_{k}}}\;\|u(t_{0})\|_{L^{p_{0}}}^{\frac{p_{0}\,\theta_{p_{0}}}{p_{k}\theta_{k}}}\leq\left[\prod_{j=1}^{k}\bigg{(}4^{j}\frac{\overline{c}}{t-t_{0}}\bigg{)}^{\frac{N(\theta_{j-1}-\theta_{j})}{\alpha}}\right]^{\frac{1}{p_{k}\theta_{k}}}\\!\\!\\!\\!\\!\\!\|u(t_{0})\|_{L^{p_{0}}}^{\frac{p_{0}\,\theta_{p_{0}}}{p_{k}\theta_{k}}}.$

Finally, letting $k\to\infty$, it is easy to see that

$\prod_{j=1}^{k}\bigg{(}4^{j}\frac{C}{t-t_{0}}\bigg{)}^{\frac{N(\theta_{j-1}-\theta_{j})}{\alpha}}\leq
2^{\frac{N}{\alpha^{2}p_{0}\,p_{k}\theta_{k}}}\left(\frac{C}{t-t_{0}}\right)^{\frac{N(\theta_{p_{0}}-\theta_{k})}{\alpha}}$

so that, using the lower semicontinuity of the $L^{\infty}$ norm, we get

$\displaystyle\|u(t)\|_{L^{\infty}}$
$\displaystyle\leq\lim\limits_{k\rightarrow\infty}\|u(t_{k})\|_{L^{p_{k}}}$
$\displaystyle\leq\lim_{k\rightarrow\infty}\left(2^{\frac{N}{\alpha^{2}{p_{0}}\,p_{k}\theta_{k}}}\left(\frac{C}{t-t_{0}}\right)^{\frac{N(\theta_{p_{0}}-\theta_{k})}{\alpha}}\right)^{\frac{1}{p_{k}\theta_{k}}}\|u(t_{0})\|_{L^{p_{0}}}^{\frac{p_{0}\,\theta_{p_{0}}}{p_{k}\theta_{k}}}\leq
C\frac{\|u(t_{0})\|_{L^{p_{0}}}^{\alpha
p_{0}\,\theta_{p_{0}}}}{(t-t_{0})^{N\theta_{p_{0}}}}\,.$

This proves the desired inequality (6.4) and concludes the proof of
(ii)$\Longrightarrow$(i).

(i)$\Longrightarrow$(ii). Follows by Theorem B.1 of Appendix B, which states
that $L^{1}$–$L^{\infty}$ imply $L^{p}$–$L^{q}$-smoothing effects, which in
turn imply subcritical GNS, by Proposition 6.13(a). This concludes the proof.
∎

######  Remark 6.17.

This proof holds for all $m\geq 1$, so in particular it also shows that
subcritical GNS imply smoothing also in the linear case $m=1$, providing an
alternative proof of the implication (b)$\Longrightarrow$(a) or
(c)$\Longrightarrow$(a) in Theorem 6.1. When $m\in(0,1)$, which corresponds to
the fast diffusion case, the same proof works as well, but we need to require
further integrability on the initial datum in order to perform the iteration,
as thoroughly explained in e.g. [28, Section 3] (and also [54, 32, 29] for the
local case).

Finally, we have:

###### Proof of Proposition 6.12.

By Lemma B.2 (with $\gamma=0$) of Appendix B, item (i) implies Proposition
6.13(b)(i), and hence, item (ii) holds. ∎

##  7\. Various examples

This section is devoted to study the operators whose Green functions satisfy
assumptions (2)–(2), and hence, which smoothing effects are satisfied by such
operators. As a consequence of Proposition D.3, we have:

######  Proposition 7.1.

Assume that the operator $-\mathfrak{L}$ is linear, symmetric, nonnegative,
and moreover, densely defined, $\mathfrak{m}$-accretive, and Dirichlet in
$L^{1}(\mathbb{R}^{N})$. Then (2) holds, and the Green functions of
$-\mathfrak{L}$ and $I-\mathfrak{L}$ are respectively given by

$\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)=\int_{0}^{\infty}\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(x,t)\,\mathrm{d}t$

and

$\mathbb{G}_{I-\mathfrak{L}}^{x_{0}}(x)=\int_{0}^{\infty}\textup{e}^{-t}\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(x,t)\,\mathrm{d}t,$

where $\mathbb{H}_{-\mathfrak{L}}^{x_{0}}$ is the corresponding heat kernel of
$-\mathfrak{L}$.

Let us illustrate these formulas through Fourier analysis. Denote by
$\sigma_{-\mathfrak{L}}$ the Fourier symbol of the operator $-\mathfrak{L}$.
Then the heat kernel can be expressed as

$\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(x,t)=\mathcal{F}^{-1}\big{[}\textup{e}^{-\sigma_{-\mathfrak{L}}(\cdot)t}\big{]}(x-x_{0})=\int_{\mathbb{R}^{N}}\textup{e}^{-\sigma_{-\mathfrak{L}}(\xi)t}\textup{e}^{2\pi\textup{i}(x-x_{0})\cdot\xi}\,\mathrm{d}\xi,$

and

$\begin{split}\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)&=\int_{0}^{\infty}\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(x,t)\,\mathrm{d}t=\int_{\mathbb{R}^{N}}\bigg{(}\int_{0}^{\infty}\textup{e}^{-\sigma_{-\mathfrak{L}}(\xi)t}\,\mathrm{d}t\bigg{)}\textup{e}^{2\pi\textup{i}(x-x_{0})\cdot\xi}\,\mathrm{d}\xi\\\
&=\int_{\mathbb{R}^{N}}\frac{1}{\sigma_{-\mathfrak{L}}(\xi)}\textup{e}^{2\pi\textup{i}(x-x_{0})\cdot\xi}\,\mathrm{d}\xi=\mathcal{F}^{-1}\Big{[}\frac{1}{\sigma_{-\mathfrak{L}}(\xi)}\Big{]}(x-x_{0}).\end{split}$

We also refer to the well-written book [17], which provides many examples of
Green functions and a good introduction to potential theory.

In the examples that follows, we will need

(7.1)
$\int_{0}^{\infty}\textup{e}^{-tr}r^{\vartheta}\,\mathrm{d}r=\frac{\Gamma(\vartheta+1)}{t^{\vartheta+1}}<\infty\qquad\text{whenever
$\vartheta>-1$},$

where $\Gamma$ is the gamma function.

###  7.1. On the assumption (2)

As demonstrated in Theorem 3.1, assumption (2) leads to the estimate

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\lesssim
t^{-N\theta_{\alpha}}\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}^{\alpha\theta_{\alpha}}\qquad\text{for
a.e. $t>0$}$

for weak dual solutions of (GPME) with initial data $u_{0}$. Let us provide
some concrete examples of operators $-\mathfrak{L}$ in (GPME) whose Green
functions satisfy (2).

######  Lemma 7.2.

The fractional Laplacian/Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ with
$\alpha\in(0,2]$ has a Green function which satisfies (2).

######  Remark 7.3.

Let us mention that heat kernel estimates for the Laplacian and the fractional
Laplacian dates back to Fourier [73, Chapter IX Section II] (see also [71,
Section 2.3]) and Blumenthal and Getoor [16], respectively.

###### Proof of Lemma 7.2.

Assume $\alpha\in(0,2)$. By Lemma 2 in Chapter V.1 in [106],

$\mathbb{G}_{(-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}(x)=\mathscr{F}^{-1}\big{[}|\cdot|^{-\alpha}\big{]}(x-x_{0})\eqsim|x-x_{0}|^{-(N-\alpha)}.$

Now,

$\int_{B_{R}(x_{0})}\mathbb{G}_{(-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}(x)\,\mathrm{d}x\eqsim\int_{0}^{R}r^{-(N-\alpha)}r^{N-1}\,\mathrm{d}r\eqsim
R^{\alpha}.$

Moreover, for any $x\in\mathbb{R}^{N}\setminus B_{R}(x_{0})$,

$\mathbb{G}_{(-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}(x)\lesssim
R^{-(N-\alpha)},$

and the result follows.

Assume $\alpha=2$. The result is classical and can e.g. be found in [55,
Section 1.1.8]. We get that
$\mathbb{G}_{-\Delta}^{x_{0}}(x)\eqsim|x-x_{0}|^{-(N-2)}$ which satisfies (2)
with $\alpha=2$. ∎

######  Corollary 7.4.

Any operator $\mathfrak{L}$ whose Green function satisfies

$\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\lesssim|x-x_{0}|^{-(N-\alpha)}\qquad\text{for
some $\alpha\in(0,2]$}$

will fulfil (2).

######  Remark 7.5.

By Lemma 6.9, the above assumption on the Green function implies that the
corresponding operator satisfies the Sobolev inequality. Again, we also refer
to [88] for a further discussion.

######  Lemma 7.6.

Assume that the real matrix $[a_{ij}]_{i,j=1,\ldots,N}$ is nonnegative and
symmetric and $\mathfrak{L}=\sum_{i,j=1}^{N}a_{ij}\partial_{x_{i}x_{j}}^{2}$.
Given the following statements:

  1. (i)

There exist constants $C,c>0$ such that

$c|y|^{2}\leq\sum_{i,j=1}^{N}a_{ij}y_{i}y_{j}\leq C|y|^{2}.$

  2. (ii)

There exist constants $C,c>0$ such that

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
ct^{-\frac{N}{2}}\textup{exp}\Big{(}-C\frac{|x-x_{0}|^{2}}{t}\Big{)}.$

  3. (iii)

There exists a constant $C>0$ such that

$\mathbb{G}_{-\mathcal{L}^{\mu}}^{x_{0}}(x)\leq C|x-x_{0}|^{-(N-2)}.$

We have (i)$\Longrightarrow$(ii)$\Longrightarrow$(iii).

######  Remark 7.7.

  1. (a)

The heat kernel bound is (up to constants) the same as for the regular
Laplacian. This is not surprising in the constant coefficient case since the
operator is, up to a translation, the Laplacian. The more interesting case is
of course when the coefficients are $(x,t)$-dependent, see also [83, Section
2.9] and the classical [3]. For a similar result in the fractional setting, we
refer to [89].

  2. (b)

The upper bound in statement (i) might seem superfluous, but the constant
inside the exponential function in (ii) depends on it.

###### Proof of Lemma 7.6.

(i)$\Longrightarrow$(ii). Follows by [55, Corollary 3.2.8] (see also [3]).

(ii)$\Longrightarrow$(iii). Follows by [55, Theorem 3.1.1] (see also [3]). ∎

The nonlocal counterpart is somehow when the Lévy measure is comparable to the
measure of the fractional Laplacian.

######  Lemma 7.8.

Assume $\mathfrak{L}=\mathcal{L}^{\mu}$ and (1). Given the following
statements, for $\alpha\in(0,2)$:

  1. (i)

There exist constants $C,c>0$ such that

$\frac{c}{|z|^{N+\alpha}}\leq\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\leq\frac{C}{|z|^{N+\alpha}}.$

  2. (ii)

There exists a constant $C>0$ such that

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
C\min\big{\\{}t^{-\frac{N}{\alpha}},t|x-x_{0}|^{-(N+\alpha)}\big{\\}}.$

  3. (iii)

There exists a constant $C>0$ such that

$\mathbb{G}_{-\mathcal{L}^{\mu}}^{x_{0}}(x)\leq C|x-x_{0}|^{-(N-\alpha)}.$

We have (i)$\Longrightarrow$(ii)$\Longrightarrow$(iii).

######  Remark 7.9.

We can also slightly weaken the assumption on the lower bound: There exist
constants $C,c>0$ such that

$c\varepsilon^{-\alpha}\leq\int_{|z|>\varepsilon}\,\mathrm{d}\mu(z)\quad\text{$\forall\,\varepsilon>0$}\qquad\text{and}\qquad\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\leq\frac{C}{|z|^{N+\alpha}}.$

The estimates on the heat kernel and Green function still hold [108, Theorem
2] with $f(x,y)=f(|y-x|)=\,\mathrm{d}\mu/\,\mathrm{d}z$, see also the recent
[88].

###### Proof of Lemma 7.8.

(i)$\Longrightarrow$(ii). Follows by [46, Theorem 1.2] with $\rho(x,y)=|x-y|$,
$V(r)=r^{N}$, $\gamma_{1}=\gamma_{2}=0$, $\psi(r)=1$, and
$\phi_{1}(r)=r^{\alpha}$.

(ii)$\Longrightarrow$(iii). By direct computations,

$\begin{split}\mathbb{G}_{-\mathcal{L}^{\mu}}^{x_{0}}(x)&=\int_{0}^{\infty}\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\,\mathrm{d}t\\\
&\lesssim\int_{0}^{|x-x_{0}|^{\alpha}}t|x-x_{0}|^{-(N+\alpha)}\,\mathrm{d}t+\int_{|x-x_{0}|^{\alpha}}^{\infty}t^{-\frac{N}{\alpha}}\,\mathrm{d}t\\\
&\lesssim|x-x_{0}|^{-(N-\alpha)}.\qed\end{split}$

###  7.2. Combinations of assumption (2)

Sometimes the Green function has different power behaviours at zero and at
infinity. As demonstrated in Theorem 3.3, such a case leads to the estimate

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\lesssim
t^{-N\theta_{\alpha}}\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}^{\alpha\theta_{\alpha}}+t^{-N\theta_{2}}\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}^{2\theta_{2}}\qquad\text{for
a.e. $t>0$}$

for weak dual solutions of (GPME) with initial data $u_{0}$. Let us provide
some concrete examples of operators $-\mathfrak{L}$ in (GPME) whose Green
functions satisfy combinations of (2).

We start with one of the most basic operators giving such an estimate:

######  Lemma 7.10.

Assume $\alpha\in(0,2)$ and
$-\mathfrak{L}=(-\Delta)+(-\Delta)^{\frac{\alpha}{2}}=:(-\Delta)+(-\mathcal{L}^{\mu})$.
Given the following statements:

  1. (i)

For some constant $C>0$, we consider

$\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)=\frac{C}{|z|^{N+\alpha}}.$

  2. (ii)

There exists a constant $C>0$ such that

$\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(x,t)\leq
C\begin{cases}f(x-x_{0},t)+\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}(x,t)\qquad\qquad&\text{if
$0<t<|x-x_{0}|^{2}\leq 1$,}\\\
\mathbb{H}_{-\Delta}^{x_{0}}(x,t)\qquad\qquad&\text{if
$|x-x_{0}|^{2}<t<|x-x_{0}|^{\alpha}\leq 1,$}\\\
\mathbb{H}_{-\Delta}^{x_{0}}(x,t)\qquad\qquad&\text{if $|x-x_{0}|^{\alpha}\leq
t\leq 1$,}\\\
\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}(x,t)\qquad\qquad&\text{if
$t\geq 1$ or $|x-x_{0}|\geq 1$,}\end{cases}$

where $f(x-x_{0},t):=(4\pi t)^{-N/2}\textup{exp}(-|x-x_{0}|^{2}/(16t))$.

  3. (iii)

There exists a constant $C>0$ such that

$\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\leq
C\begin{cases}|x-x_{0}|^{-(N-2)}\qquad\qquad&\text{if $|x|\leq 1$,}\\\
|x-x_{0}|^{-(N-\alpha)}\qquad\qquad&\text{if $|x|>1$.}\\\ \end{cases}$

We have (i)$\Longrightarrow$(ii)$\Longrightarrow$(iii).

######  Remark 7.11.

Note that when $0<R\leq 1$, we get

$\int_{B_{R}(x_{0})}\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x\lesssim
R^{2},\qquad\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\lesssim
R^{-(N-2)}\quad\text{for $x\in\mathbb{R}^{N}\setminus B_{R}(x_{0})$,}$

and when $R>1$,

$\int_{B_{R}(x_{0})}\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x\lesssim
R^{\alpha},\qquad\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\lesssim
R^{-(N-\alpha)}\quad\text{for $x\in\mathbb{R}^{N}\setminus B_{R}(x_{0})$.}$

Hence, we are in the setting of Theorem 3.3 although in this example the small
time behaviour is governed by the Laplacian and the large time by the
fractional Laplacian.

###### Proof of Lemma 7.10.

(i)$\Longrightarrow$(ii). Follows by [105, Theorem 2.13].

(ii)$\Longrightarrow$(iii). Assume $|x|\leq 1$. Then

$\begin{split}\mathbb{G}_{-\mathfrak{L}}^{0}(x)&=\int_{0}^{\infty}\mathbb{H}_{-\mathfrak{L}}^{0}(x,t)\,\mathrm{d}t=\bigg{(}\int_{0}^{|x|^{2}}+\int_{|x|^{2}}^{|x|^{\alpha}}+\int_{|x|^{\alpha}}^{1}+\int_{1}^{\infty}\bigg{)}\mathbb{H}_{-\mathfrak{L}}^{0}(x,t)\,\mathrm{d}t.\end{split}$

Let us start with the integral involving $f(x,t)$. The change of variables
$|x|^{2}/(16t)\mapsto t$ gives

$\begin{split}\int_{0}^{|x|^{2}}f(x,t)\,\mathrm{d}t\eqsim\int_{0}^{|x|^{2}}t^{-N/2}\textup{e}^{\frac{-|x|^{2}}{16t}}\,\mathrm{d}t=\Big{(}\frac{|x|^{2}}{16}\Big{)}^{-\frac{N}{2}+1}\int_{\frac{1}{16}}^{\infty}\tau^{\frac{N}{2}-2}\textup{e}^{-\tau}\,\mathrm{d}\tau\eqsim|x|^{-(N-2)},\end{split}$

where we estimated the final integral by (7.1). The integrals involving
$\mathbb{H}_{-\Delta}^{0}$ can be estimated in a similar way. It remains to
estimate the contribution from
$\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}$:

$\begin{split}&\bigg{(}\int_{0}^{|x|^{2}}+\int_{1}^{\infty}\bigg{)}\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(x,t)\,\mathrm{d}t\lesssim\bigg{(}\int_{0}^{|x|^{2}}+\int_{1}^{\infty}\bigg{)}\min\\{t^{-N/\alpha},t|x|^{-N-\alpha}\\}\,\mathrm{d}t\\\
&=\int_{0}^{|x|^{2}}t|x|^{-N-\alpha}\,\mathrm{d}t+\int_{1}^{\infty}t^{-N/\alpha}\,\mathrm{d}t\eqsim|x|^{-(N-2)+2-\alpha}+1.\end{split}$

Since $|x|\leq 1$, we have $|x|^{-(N-2)+2-\alpha}\leq|x|^{-(N-2)}$ and
$1\leq|x|^{-(N-2)}$.

Assume $|x|>1$. Then

$\begin{split}\mathbb{G}_{-\mathfrak{L}}^{0}(x)&=\int_{0}^{\infty}\mathbb{H}_{-\mathfrak{L}}^{0}(x,t)\,\mathrm{d}t\lesssim\int_{0}^{\infty}\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(x,t)\,\mathrm{d}t\lesssim|x|^{-(N-\alpha)}.\end{split}$

We combine the results to complete the proof. ∎

Let us now consider the relativistic Schrödinger type operators like

(7.2)
$(\kappa^{2}I-\Delta)^{\frac{\alpha}{2}}-\kappa^{\alpha}I\qquad\text{with
$\kappa>0$ and $\alpha\in(0,2)$}$

are Lévy operators [100, Lemma 2] (see also [79] and [72, Appendix B]), i.e.,
they can be written on the form $\mathcal{L}^{\mu}$, see (1.1), with a measure
satisfying (1).

######  Lemma 7.12.

Assume $-\mathfrak{L}$ is given by (7.2). Given the following statements, for
$\alpha\in(0,2)$:

  1. (i)

For some constant $C>0$, we consider

$\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)=\frac{C}{|z|^{\frac{N+\alpha}{2}}}K_{\frac{N+\alpha}{2}}(\kappa|z|),$

where $K_{a}$ is the modified Bessel function of the second kind with index
$a\in\mathbb{R}$.

  2. (ii)

There exists a constant $C>0$ such that, for all $\gamma>2-\alpha$,

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
C\min\bigg{\\{}t^{-\frac{N}{\alpha}},\frac{t}{|x-x_{0}|^{N+\alpha}(1+|x-x_{0}|)^{\gamma}}\bigg{\\}}\qquad\text{if
$0<t<1$,}$

and

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
C\min\bigg{\\{}t^{-\frac{N}{2}},\frac{t}{|x-x_{0}|^{N+\alpha}(1+|x-x_{0}|)^{\gamma}}\bigg{\\}}\qquad\text{if
$t>1$.}$

  3. (iii)

There exists a constant $C>0$ such that

$\mathbb{G}_{-\mathcal{L}^{\mu}}^{x_{0}}(x)\leq
C\big{(}|x-x_{0}|^{-(N-\alpha)}+|x-x_{0}|^{-(N-2)}\big{)}.$

We have (i)$\Longrightarrow$(ii)$\Longrightarrow$(iii).

######  Remark 7.13.

It is interesting to note that

$\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\eqsim\frac{1}{|z|^{N+\alpha}}\quad\text{as
$|z|\to
0$}\qquad\text{and}\qquad\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\eqsim\frac{1}{|z|^{\frac{N+\alpha+1}{2}}}\textup{e}^{-\kappa|z|}\quad\text{as
$|z|\to\infty$.}$

###### Proof of Lemma 7.12.

(i)$\Longrightarrow$(ii). Follows by [109, Section 5].

(ii)$\Longrightarrow$(iii). Since

$\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x)=\int_{0}^{\infty}\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(x,t)\,\mathrm{d}t,$

we will have to consider three cases

${\rm(I)}\qquad\frac{t}{|x|^{N+\alpha}(1+|x|)^{\gamma}}>t\qquad\Longleftrightarrow\qquad|x|^{N+\alpha}(1+|x|)^{\gamma}<1,$
${\rm(II)}\qquad\frac{t}{|x|^{N+\alpha}(1+|x|)^{\gamma}}=t\qquad\Longleftrightarrow\qquad|x|^{N+\alpha}(1+|x|)^{\gamma}=1,$

and

${\rm(III)}\qquad\frac{t}{|x|^{N+\alpha}(1+|x|)^{\gamma}}<t\qquad\Longleftrightarrow\qquad|x|^{N+\alpha}(1+|x|)^{\gamma}>1.$

In the case of (I), we have three different behaviours:

$\begin{split}\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x)&\eqsim\int_{0}^{|x|^{\alpha}(1+|x|)^{\gamma\frac{\alpha}{N+\alpha}}}\frac{t}{|x|^{N+\alpha}(1+|x|)^{\gamma}}\,\mathrm{d}t+\int_{|x|^{\alpha}(1+|x|)^{\gamma\frac{\alpha}{N+\alpha}}}^{1}t^{-\frac{N}{\alpha}}\,\mathrm{d}t+\int_{1}^{\infty}t^{-\frac{N}{2}}\,\mathrm{d}t\\\
&=\frac{1}{2}|x|^{-(N-\alpha)}(1+|x|)^{-\gamma\frac{N-\alpha}{N+\alpha}}+\frac{\alpha}{N-\alpha}|x|^{-(N-\alpha)}(1+|x|)^{-\gamma\frac{N-\alpha}{N+\alpha}}-\frac{\alpha}{N-\alpha}+\frac{2}{N-2}\\\
&=\frac{1}{2}\frac{N-\alpha}{N+\alpha}|x|^{-(N-\alpha)}(1+|x|)^{-\gamma\frac{N-\alpha}{N+\alpha}}+\frac{N(2-\alpha)}{(N-2)(N-\alpha)}\\\
&\leq\Big{(}\frac{1}{2}\frac{N-\alpha}{N+\alpha}+\frac{N(2-\alpha)}{(N-2)(N-\alpha)}\Big{)}|x|^{-(N-\alpha)}\qquad\text{in
$\\{x\,:\,|x|^{N+\alpha}(1+|x|)^{\gamma}<1\\}$,}\end{split}$

were we used $1+|x|\geq 1$ to get

$\frac{1}{(1+|x|)^{\gamma\frac{N-\alpha}{N+\alpha}}}\leq 1.$

In the case of (II), we have two different behaviours:

$\begin{split}\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x)&\eqsim\int_{0}^{1}t\,\mathrm{d}t+\int_{1}^{\infty}t^{-\frac{N}{2}}\,\mathrm{d}t=\frac{1}{2}+\frac{2}{N-2}=\frac{1}{2}\frac{N-2}{N+2}.\end{split}$

In the case of (III), we have two different behaviours:

$\begin{split}\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x)&\eqsim\int_{0}^{|x|^{(N+\alpha)\frac{2}{N+2}}(1+|x|)^{\gamma\frac{2}{N+2}}}\frac{t}{|x|^{N+\alpha}(1+|x|)^{\gamma}}\,\mathrm{d}t+\int_{|x|^{(N+\alpha)\frac{2}{N+2}}(1+|x|)^{\gamma\frac{2}{N+2}}}^{\infty}t^{-\frac{N}{2}}\,\mathrm{d}t\\\
&=\frac{1}{2}|x|^{-(N-2)}\Big{(}\frac{|x|^{2-\alpha}}{(1+|x|)^{\gamma}}\Big{)}^{\frac{N-2}{N+2}}+\frac{2}{N-2}|x|^{-(N-2)}\Big{(}\frac{|x|^{2-\alpha}}{(1+|x|)^{\gamma}}\Big{)}^{\frac{N-2}{N+2}}\\\
&=\frac{1}{2}\frac{N-2}{N+2}|x|^{-(N-2)}\Big{(}\frac{|x|^{2-\alpha}}{(1+|x|)^{\gamma}}\Big{)}^{\frac{N-2}{N+2}}.\end{split}$

By the assumption $\gamma>2-\alpha$, we get

$\begin{split}\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x)\lesssim|x|^{-(N-2)}\qquad\text{in
$\\{x\,:\,|x|^{N+\alpha}(1+|x|)^{\gamma}>1\\}$.}\end{split}$

We then collect the three cases in one estimate to complete the proof. ∎

Finally, we also consider the generator of a finite range isotropically
symmetric $\alpha$-stable process in $\mathbb{R}^{N}$ with jumps of size
larger than $1$ removed.

######  Lemma 7.14.

Assume $\mathfrak{L}=\mathcal{L}^{\mu}$ with a measure $\mu$ satisfying (1).
Given the following statements, for $\alpha\in(0,2)$:

  1. (i)

There exists a constant $C>0$ such that

$\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)=\frac{C}{|z|^{N+\alpha}}\mathbf{1}_{|z|\leq
1}.$

  2. (ii)

There exist constants $C,c>0$ and $0<C_{*},R_{*}<1$ such that,

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
C\min\big{\\{}t^{-\frac{N}{\alpha}},t|x-x_{0}|^{-(N+\alpha)}\big{\\}},\qquad\text{$0<t<R_{*}^{\alpha}$,
$|x-x_{0}|\leq R_{*}$,}$ $\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
C\exp\Big{(}-c|x-x_{0}|\log\Big{(}\frac{|x-x_{0}|}{t}\Big{)}\Big{)},\qquad\text{$|x-x_{0}|\geq\max\\{t/C_{*},R_{*}\\}$,}$

and

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
Ct^{-\frac{N}{2}}\exp\Big{(}-c\frac{|x-x_{0}|^{2}}{t}\Big{)},\qquad\text{$t>R_{*}^{\alpha}$,
$|x-x_{0}|\leq t/C_{*}$.}$

  3. (iii)

There exists a constant $C>0$ such that

$\mathbb{G}_{-\mathcal{L}^{\mu}}^{x_{0}}(x)\leq
C\big{(}|x-x_{0}|^{-(N-\alpha)}+|x-x_{0}|^{-(N-2)}\big{)}.$

We have (i)$\Longrightarrow$(ii)$\Longrightarrow$(iii).

######  Remark 7.15.

We are again in the setting of Theorem 3.3 by Remark 7.11.

###### Proof of Lemma 7.14.

(i)$\Longrightarrow$(ii). Follows by [44, Proposition 2.1 and Theorem 2.3].

(ii)$\Longrightarrow$(iii). Follows by the proof of [44, Theorem 4.7]. ∎

###  7.3. On the assumption (2)

If the Green function decays fast enough at infinity, the function itself will
not only be $L_{\textup{loc}}^{1}$ but indeed $L^{1}$, see (2). As
demonstrated in Theorem 3.7, such a case leads to the estimate

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\lesssim
t^{-1/(m-1)}\qquad\text{for a.e. $t>0$},$

for weak dual solutions of (GPME) with initial data $u_{0}$. Let us provide
some concrete examples of operators $-\mathfrak{L}$ in (GPME) whose Green
functions satisfy (2). Actually, any operator of the form $I-\mathfrak{L}$ has
a Green function satisfying (2). In what follows, we will explain this result,
and illustrate it with other examples as well.

######  Lemma 7.16.

Under the assumptions of Proposition 7.1,

$\|\mathbb{G}_{I-\mathfrak{L}}^{x_{0}}\|_{L^{1}(\mathbb{R}^{N})}=\|\mathbb{G}_{I-\mathfrak{L}}^{0}\|_{L^{1}(\mathbb{R}^{N})}\leq
1.$

Hence, the operator $I-\mathfrak{L}$ has a Green function which satisfies (2).

###### Proof.

Assumption (2), an application of the Tonelli lemma, and the fact that, by
Remark 4.2 (i.e., decay of $L^{1}$-norm),
$\int\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(\cdot,t)=\int\mathbb{H}_{-\mathfrak{L}}^{0}(\cdot,t)\leq
1$ for every fixed $t>0$ concludes the proof. ∎

######  Remark 7.17.

We immediately see that the presence of the identity operator is crucial. In
fact, if $-\mathfrak{L}$ is such that the corresponding heat equation
preserves mass, then
$\|\mathbb{G}_{-\mathfrak{L}}^{x_{0}}\|_{L^{1}(\mathbb{R}^{N})}=\infty$ (cf.
Proposition 7.1). Examples of mass preserving operators are Lévy operators
(1.1) with $c=0$.

Let us begin by considering the extreme case $-\mathfrak{L}=I$ for which the
PDE in (GPME) reads

$\partial_{t}u=-u^{m}.$

For any function $t\mapsto Y(t)$, that equation is an ODE of the form

$Y^{\prime}(t)=-Y(t)^{1+(m-1)}\quad\Longrightarrow\quad
Y(t)\leq\Big{(}\frac{1}{(m-1)t}\Big{)}^{\frac{1}{m-1}}.$

Hence, by the comparison principle for (GPME) with $-\mathfrak{L}=I$ (where we
take $Y(0)=\infty$), we get the absolute bound

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq
Y(t)\leq\Big{(}\frac{1}{(m-1)t}\Big{)}^{\frac{1}{m-1}}.$

See also Section III.C in [117]. The above is also contained in the following
lemma:

######  Lemma 7.18.

The identity operator $-\mathfrak{L}=I$ has a Green function which satisfies
(2), i.e.,

$\|\mathbb{G}_{I}^{x_{0}}\|_{L^{1}(\mathbb{R}^{N})}=\|\mathbb{G}_{I}^{0}\|_{L^{1}(\mathbb{R}^{N})}\leq
C_{1}<\infty.$

###### Proof.

We obtain

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{\mathbb{R}^{N}}\mathbb{H}_{I}^{x_{0}}(x,t)\,\mathrm{d}x=-\int_{\mathbb{R}^{N}}\mathbb{H}_{I}^{x_{0}}(x,t)\,\mathrm{d}x,$

i.e., $\int\mathbb{H}_{I}^{x_{0}}(\cdot,t)=\textup{e}^{-t}$ for all $t>0$. By
the definition of the Green function (cf. Proposition 7.1), we conclude. ∎

######  Remark 7.19.

  1. (a)

The proof also demonstrates that the operator $-\mathfrak{L}=I$ is not
conserving mass. Indeed, in the corresponding heat equation, it decays with
time.

  2. (b)

Moreover, it provides a trivial example of an operator which yields
boundedness in the nonlinear case $(m>1)$, but not in the linear case ($m=1$).

Of course, we can reapply the same strategy of comparison with $Y(t)$ for
(GPME) with $-\mathfrak{L}\mapsto I-\mathfrak{L}$ to get

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq
Y(t)\leq\Big{(}\frac{1}{(m-1)t}\Big{)}^{\frac{1}{m-1}}$

independently of $\mathfrak{L}$!

######  Remark 7.20.

When $m=1$, we need to adapt another strategy (since
$Y(t)=Y(0)\textup{e}^{-t})$, but recall that by defining

$u(x,t):=\textup{e}^{-t}v(x,t)$

where $v$ solves (GPME) with $m=1$, then $u$ solves

$\begin{cases}\partial_{t}u-\mathfrak{L}[u]+u=0\qquad\qquad&\text{in}\qquad\mathbb{R}^{N}\times(0,T],\\\
u(\cdot,0)=u_{0}\qquad\qquad&\text{on}\qquad\mathbb{R}^{N}.\end{cases}$

Hence, in this case, $L^{1}$–$L^{\infty}$-smoothing follows as long as it
holds for $v$, i.e., as long as $\mathfrak{L}$ is strong enough to provide it.

When $\kappa=1$ in (7.2), we get the Lévy operator

$-\mathcal{L}^{\mu_{\text{RS}}}:=(I-\Delta)^{\frac{\alpha}{2}}-I,$

i.e., $(I-\Delta)^{\frac{\alpha}{2}}$ is of the form
$(I-\mathcal{L}^{\mu_{\text{RS}}})$. Hence:

######  Lemma 7.21.

The operator $-\mathfrak{L}=(I-\Delta)^{\frac{\alpha}{2}}$ with
$\alpha\in(0,2)$ has a Green function which satisfies (2), i.e.,

$\|\mathbb{G}_{(I-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}\|_{L^{1}(\mathbb{R}^{N})}=\|\mathbb{G}_{(I-\Delta)^{\frac{\alpha}{2}}}^{0}\|_{L^{1}(\mathbb{R}^{N})}\leq
C_{1}<\infty.$

######  Remark 7.22.

The operator $-\mathfrak{L}=(I-\Delta)^{\frac{\alpha}{2}}$ with $\alpha=1$
appears e.g. in [6] for the linear equation (GPME) with $m=1$. Since the
mentioned operator is of the form $(I-\mathcal{L}^{\mu_{\text{RS}}})$,
$L^{1}$–$L^{\infty}$-smoothing holds whenever it holds for
$\mathcal{L}^{\mu_{\text{RS}}}$ (see Remark 7.20). The heat kernel bounds of
Lemma 7.12 then provides the result through Theorem 6.1. In the nonlinear case
($m>1$), however, the above lemma ensures that (2) holds and we deduce
absolute bounds.

###### Proof of Lemma 7.21.

Note that

$\mathbb{G}_{(I-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}(x)=\mathscr{F}^{-1}\big{[}(1+|\cdot|^{2})^{-\frac{\alpha}{2}}\big{]}(x-x_{0}),$

i.e., the Bessel potential. The result then follows by Proposition 2 in
Chapter V.3 in [106]. Or, we can simply note that
$\mathbb{G}_{(I-\Delta)^{\alpha/2}}^{x_{0}}=\mathbb{G}_{I-\mathcal{L}^{\mu_{\text{RS}}}}^{x_{0}}$
and $-\mathcal{L}^{\mu_{\text{RS}}}$ is such that the heat equation has
$L^{1}$-decay, cf. Lemma 7.16. ∎

Again, the operator related to the Bessel potential does not conserve mass
since $(I-\Delta)^{\frac{\alpha}{2}}-I$ does so. Moreover, in the latter case,
assumption (2) cannot hold (cf. Remark 7.17) and we can write the PDE in
(GPME) as

$\partial_{t}u+(I-\Delta)^{\frac{\alpha}{2}}[u^{m}]=u^{m}.$

This is in contrast with the operators $I-\mathfrak{L}$ in Lemma 7.16 which
satisfies the PDE

$\partial_{t}u-\mathfrak{L}[u^{m}]=-u^{m}.$

Taking $-\mathfrak{L}=(I-\Delta)^{\frac{\alpha}{2}}$ in the latter, we see
that assumption (2) relies on either the (strong) absorption term being
present or the operator itself being _positive_ , or also both being present
of course.

###  7.4. On the assumption (2)

By Corollary D.6, $\mathbb{G}_{I-\mathfrak{L}}^{x_{0}}$ has at least as good
integrability properties as $\mathbb{G}_{-\mathfrak{L}}^{x_{0}}$. It is,
moreover, always defined for descent operators $-\mathfrak{L}$, see the
discussion in Section D. There should therefore be no surprise that assumption
(2) is quite general, however, as shown in Theorem 3.5, it provides a rather
poor smoothing estimate:

(7.3) $\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\lesssim
t^{-\frac{1}{m-1}}+\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}\qquad\text{for a.e.
$t>0$,}$

for weak dual solutions of (GPME) with initial data $u_{0}$. Let us provide
some concrete examples of operators $-\mathfrak{L}$ in (GPME) whose Green
functions satisfy (2), and let us also see how to improve the above estimate.
To continue, we advice the reader to recall (4.5).

######  Lemma 7.23.

The fractional Laplacian/Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ with
$\alpha\in(0,2]$ has a Green function which satisfies (2), i.e.,

$\|\mathbb{G}_{I+(-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}\|_{L^{p}(\mathbb{R}^{N})}=\|\mathbb{G}_{I+(-\Delta)^{\frac{\alpha}{2}}}^{0}\|_{L^{p}(\mathbb{R}^{N})}\leq
C_{p}<\infty$

for some $p\in(1,N/(N-\alpha))$.

###### Proof.

We use (7.1) with $\vartheta:=\frac{N}{\alpha}-1$ to obtain

(7.4)
$\begin{split}&\|\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}=\|\mathscr{F}^{-1}[\textup{e}^{-t|\xi|^{\alpha}}]\|_{L^{\infty}(\mathbb{R}^{N})}\leq\int_{\mathbb{R}^{N}}\textup{e}^{-t|\xi|^{\alpha}}\,\mathrm{d}\xi\\\
&\eqsim\int_{0}^{\infty}\textup{e}^{-tr^{\alpha}}r^{N-1}\,\mathrm{d}r=\frac{1}{\alpha}\int_{0}^{\infty}\textup{e}^{-tr}r^{\vartheta}\,\mathrm{d}r=\frac{1}{\alpha}t^{-\frac{N}{\alpha}}\Gamma\Big{(}\frac{N}{\alpha}\Big{)},\end{split}$

and hence, by Remark 4.2 (i.e., $L^{1}$-decay for $m=1$),

$\begin{split}\|\mathbb{G}_{I+(-\Delta)^{\frac{\alpha}{2}}}^{0}\|_{L^{p}(\mathbb{R}^{N})}&\leq\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\,\mathrm{d}t\\\
&\leq\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}^{\frac{p-1}{p}}\|\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(\cdot,t)\|_{L^{1}(\mathbb{R}^{N})}^{\frac{1}{p}}\,\mathrm{d}t\\\
&\lesssim\int_{0}^{\infty}\textup{e}^{-t}t^{-\frac{N}{\alpha}\frac{p-1}{p}}\,\mathrm{d}t=\tau^{-\frac{N}{\alpha}\frac{p-1}{p}+1}\int_{0}^{\infty}\textup{e}^{-\tau
r}r^{-\frac{N}{\alpha}\frac{p-1}{p}}\,\mathrm{d}r,\end{split}$

which is finite if $p<N/(N-\alpha)$ due to (7.1) again. ∎

The fractional Laplacian, however, satisfies our strongest assumption (2) as
well. We will therefore consider an operator which satisfies (2), but for
which it is not possible to verify (2) or (2). To that end, consider the sum
of onedimensional fractional Laplacians:

(7.5)
$-\mathfrak{L}=\sum_{i=1}^{N}(-\partial_{x_{i}x_{i}}^{2})^{\frac{\alpha_{i}}{2}}\qquad\text{with
$\alpha_{i}\in(0,2)$.}$

It can be written on the form $\mathcal{L}^{\mu}$ with $\mu$, for some
constant $C>0$, given by

$\,\mathrm{d}\mu(z)=C\sum_{i=1}^{N}\frac{1}{|z_{i}|^{1+\alpha_{i}}}\,\mathrm{d}z_{i}\prod_{j\neq
i}\,\mathrm{d}\delta_{0}(z_{j}).$

This measure satisfies (1) since each onedimensional fractional Laplacian
measure does, and we have:

######  Lemma 7.24.

Assume $\mathfrak{L}$ is given by (7.5) and

$\sum_{i=1}^{N}\frac{1}{\alpha_{i}}>1\qquad\text{where $\alpha_{i}\in(0,2)$}.$

Then

$\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{x_{0}}\|_{L^{p}(\mathbb{R}^{N})}=\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}\leq
C_{p}<\infty$

for some

$p\in\Big{(}1,\frac{\sum_{i=1}^{N}\frac{1}{\alpha_{i}}}{\sum_{i=1}^{N}\frac{1}{\alpha_{i}}-1}\Big{)}.$

######  Remark 7.25.

  1. (a)

Note that if $\alpha_{i}=\alpha$ for all $i\in\\{1,\ldots,N\\}$, then

$\sum_{i=1}^{N}\frac{1}{\alpha_{i}}>1\qquad\Longrightarrow\qquad\frac{N}{\alpha}>1.$

We thus recover the condition of Lemma 7.23.

  2. (b)

Various extensions within the framework of anisotropic fractional Laplacians
can be found in [19, 110, 118, 20].

  3. (c)

By [9, Section 3],

(7.6)
$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)=\prod_{i=1}^{N}\mathbb{H}_{(-\partial_{x_{i}x_{i}}^{2})^{\frac{\alpha_{i}}{2}}}^{x_{0}}(x,t)\leq
C\prod_{i=1}^{N}\rho_{i}^{x_{0,i}}(x_{i},t),$

where

$\rho_{i}^{x_{0,i}}(x_{i},t)=\min\big{\\{}t^{-\frac{1}{\alpha_{i}}},t|x_{i}-x_{0,i}|^{-(1+\alpha_{i})}\big{\\}}.$

However, the example stated at the end of [18] shows that for
$\alpha_{i}=\alpha$ with $\alpha\leq(N-1)/2<N$, small times (hence all times),
and the choice $x=\xi e_{1}$ with $\xi>0$, yields
$\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x,t)=\infty$. We thus conclude that at
least in this case, it is not possible to verify the second parts of (2) or
(2).

  4. (d)

Solutions of (GPME) with $-\mathfrak{L}$ defined by (7.5) satisfy (7.3). Once
this estimate is established, we can, moreover, use the scaling of the
operator to get it on an invariant form. We restrict to the case
$\alpha_{i}=\alpha$ for all $i\in\\{1,\ldots,N\\}$. As in Remark 3.2, if $u$
solves (GPME), then

$u_{\kappa,\Xi,\Lambda}(x,t):=\kappa u(\Xi x,\Lambda t)\qquad\text{for all
$\kappa,\Xi,\Lambda>0$}$

also solves (GPME) as long as $\kappa^{m-1}\Xi^{\alpha}=\Lambda$. This means
that

$\|u_{\kappa,\Xi,\Lambda}(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\lesssim
t^{-\frac{1}{m-1}}+\|u_{\kappa,\Xi,\Lambda}(\cdot,0)\|_{L^{1}(\mathbb{R}^{N})}$

or

$\|u(\Xi\cdot,\Lambda
t)\|_{L^{\infty}(\mathbb{R}^{N})}\lesssim\Xi^{\frac{\alpha}{m-1}}(\Lambda
t)^{-\frac{1}{m-1}}+\Xi^{-N}\|u(\cdot,0)\|_{L^{1}(\mathbb{R}^{N})}.$

The choice $\Xi=\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}^{(m-1)\theta}(\Lambda
t)^{\theta}$ and $\Lambda t\mapsto t$, then gives

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{C}{t^{N\theta}}\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}^{\alpha\theta}\qquad\text{for
a.e. $t>0$},$

where $\theta=(\alpha+N(m-1))^{-1}$ and $C$ now depending on $C_{p}$ instead
of $K_{1}$ and $K_{2}$.111111In the case $\alpha_{i}=\alpha$ for all
$i\in\\{1,\ldots,N\\}$, the bilinear form of the operator (7.5) is comparable
to the bilinear form of the fractional Laplacian, and one could instead use
the Sobolev inequality for the latter operator (see e.g. [56]) together with a
Moser iteration to obtain the $L^{1}$–$L^{\infty}$-smoothing. This in turn
implies the corresponding Nash inequality (see Section 6.2). Note that, even
in the case $\alpha_{i}\neq\alpha_{j}$, one can deduce the Sobolev inequality,
from which the Nash inequality follows, by scratch [43, Theorem 2.4]. This
will then ensure the $L^{1}$–$L^{\infty}$-smoothing estimate both in the
linear and nonlinear case.

###### Proof of Lemma 7.24.

Recall that the heat kernel is given by (7.6). Since we are considering a Lévy
operator, it provides $L^{1}$-decay, and by (7.4) with $N=1$, we get

$\begin{split}&\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}\leq\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\,\mathrm{d}t\\\
&\leq\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}^{\frac{p-1}{p}}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{1}(\mathbb{R}^{N})}^{\frac{1}{p}}\,\mathrm{d}t=\int_{0}^{\infty}\textup{e}^{-t}\bigg{\|}\prod_{i=1}^{N}\mathbb{H}_{(-\partial_{x_{i}x_{i}}^{2})^{\alpha_{i}/2}}^{0}(\cdot_{i},t)\bigg{\|}_{L^{\infty}(\mathbb{R}^{N})}^{\frac{p-1}{p}}\,\mathrm{d}t\\\
&\leq\int_{0}^{\infty}\textup{e}^{-t}\prod_{i=1}^{N}\|\mathbb{H}_{(-\partial_{x_{i}x_{i}}^{2})^{\alpha_{i}/2}}^{0}(\cdot_{i},t)\|_{L^{\infty}(\mathbb{R})}^{\frac{p-1}{p}}\,\mathrm{d}t\lesssim\int_{0}^{\infty}\textup{e}^{-t}\prod_{i=1}^{N}\bigg{(}\frac{1}{\alpha_{i}}t^{-\frac{1}{\alpha_{i}}}\Gamma\Big{(}\frac{1}{\alpha_{i}}\Big{)}\bigg{)}^{\frac{p-1}{p}}\,\mathrm{d}t\\\
&\eqsim\int_{0}^{\infty}\textup{e}^{-t}t^{-\frac{p-1}{p}\sum_{i=1}^{n}\frac{1}{\alpha_{i}}}\,\mathrm{d}t.\end{split}$

Again, by (7.1), the result follows. ∎

Note that we have exploited the fact that
$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)$ indeed has the on-diagonal
upper bound $t^{-\sum_{i=1}^{N}\frac{1}{\alpha_{i}}}$, see e.g. Corollary 3.2
in [118]. Since the off-diagonal bound cannot give a useful Green function
estimate (in all cases), we resort to our assumption (2). We refer the reader
to Remark 6.2 which provides various examples of on-diagonal bounds.

Let us turn our attention to another interesting example where only a useful
on-diagonal bound can be deduced:

######  Lemma 7.26.

Assume $\mathfrak{L}=\mathcal{L}^{\mu}$ with a measure $\mu$ satisfying (1)
and, for $\alpha\in(0,2)$ and constants $C_{1},C_{2},C_{3}>0$,

$\frac{C_{1}}{|z|^{N+\alpha}}\mathbf{1}_{|z|\leq
1}\leq\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\leq\frac{C_{2}}{|z|^{N+\alpha}}\mathbf{1}_{|z|\leq
1}\qquad\text{and}\qquad\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\leq
C_{3}\mathbf{1}_{|z|>1}.$

Then, there exists a constant $C>0$ such that

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)\leq
Ct^{-\frac{N}{\alpha}}\textup{e}^{t},$

and, moreover,

$\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{x_{0}}\|_{L^{p}(\mathbb{R}^{N})}=\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}\leq
C_{p}<\infty$

for some $p\in(1,N/(N-\alpha))$.

###### Proof.

The estimate on the heat kernel follows by the beginning of Section 2 in [45]
with $V(r)=r^{N}$ and $\phi(r)=r^{\alpha}$, and since the heat kernel is
proven to be Hölder continuous in Section 3 of the same reference (so that the
exceptional set is empty). Note that the proof uses that
$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)=\textup{e}^{t}\mathbb{H}_{I-\mathcal{L}^{\mu}}^{x_{0}}(x,t)$.
We then get

$\begin{split}\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}&\leq\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\,\mathrm{d}t\\\
&\leq\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{N})}^{\frac{p-1}{p}}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{1}(\mathbb{R}^{N})}^{\frac{1}{p}}\,\mathrm{d}t\\\
&\eqsim\int_{0}^{\infty}\textup{e}^{-t}\big{(}t^{-\frac{N}{\alpha}}\textup{e}^{t}\big{)}^{\frac{p-1}{p}}\,\mathrm{d}t=\int_{0}^{\infty}\textup{e}^{-\frac{t}{p}}t^{-\frac{N}{\alpha}\frac{p-1}{p}}\,\mathrm{d}t.\end{split}$

Again, by (7.1), the result follows. ∎

###  7.5. A nonexample of our theory

We will now consider a Lévy operator which does not satisfy any of (2)–(2).

Consider the generator of a subordinate Brownian motion with Fourier symbol
$\phi(|\xi|^{2})$ where $\phi(\lambda):=\log(1+\lambda^{\frac{\alpha}{2}})$.
This process is known as a rotationally invariant _geometric_ $\alpha$-stable
process, see Section 5 in [17].

######  Lemma 7.27.

Assume $\mathfrak{L}=\mathcal{L}^{\mu}$ with a measure $\mu$ satisfying (1).
If $\mathcal{L}^{\mu}$ has Fourier symbol given by

$\log(1+|\xi|^{\alpha}),$

then the heat kernel is given by

$\mathbb{H}_{-\mathcal{L}^{\mu}}^{x_{0}}(x,t)=\frac{1}{\Gamma(t)}\int_{0}^{\infty}\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{x_{0}}(x,s)s^{t-1}\textup{e}^{-s}\,\mathrm{d}s.$

Moreover,

$\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{x_{0}}\|_{L^{1}(\mathbb{R}^{N})}=\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{1}(\mathbb{R}^{N})}\leq
C_{1}<\infty.$

######  Remark 7.28.

The operator is indeed a Lévy operator, therefore it is not surprising that it
provides $L^{1}$-decay. It is, moreover, worth noting that Theorems 5.45 and
5.46 in [17] establish that the density of the Lévy measure corresponding to
the rotationally invariant geometric $\alpha$-stable process satisfies

$\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\eqsim\frac{1}{|z|^{N}}\quad\text{as
$|z|\to
0$}\qquad\text{and}\qquad\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)\eqsim\frac{1}{|z|^{N+\alpha}}\quad\text{as
$|z|\to\infty$}.$

In fact,

$\frac{\,\mathrm{d}\mu}{\,\mathrm{d}z}(z)=\int_{0}^{\infty}\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(z,s)s^{-1}\textup{e}^{-s}\,\mathrm{d}s,$

see equation (5.69) in [17].

###### Proof of Lemma 7.27.

The formula for the heat kernel is given by equation (5.68) in [17]. Moreover,
since $(-\Delta)^{\frac{\alpha}{2}}$ actually provides conservation of mass,
we get

$\begin{split}\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{1}(\mathbb{R}^{N})}&=\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{1}(\mathbb{R}^{N})}\,\mathrm{d}t\\\
&=\int_{0}^{\infty}\textup{e}^{-t}\frac{1}{\Gamma(t)}\int_{0}^{\infty}\|\mathbb{H}_{(-\Delta)^{\frac{\alpha}{2}}}^{0}(\cdot,s)\|_{L^{1}(\mathbb{R}^{N})}s^{t-1}\textup{e}^{-s}\,\mathrm{d}s\,\mathrm{d}t\\\
&=\int_{0}^{\infty}\textup{e}^{-t}\frac{1}{\Gamma(t)}\int_{0}^{\infty}s^{t-1}\textup{e}^{-s}\,\mathrm{d}s\,\mathrm{d}t=\int_{0}^{\infty}\textup{e}^{-t}\frac{\Gamma(t)}{\Gamma(t)}\,\mathrm{d}t=1.\qed\end{split}$

The proof also demonstrates that (2) cannot hold, and moreover, neither can
(2), (2), and (2):

######  Lemma 7.29.

Assume $\mathfrak{L}=\mathcal{L}^{\mu}$ with a measure $\mu$ satisfying (1).
If $\mathcal{L}^{\mu}$ has Fourier symbol given by

$\log(1+|\xi|^{\alpha}),$

then there is some $R>0$ such that

$\int_{B_{R}(x_{0})}\mathbb{G}_{-\mathcal{L}^{\mu}}^{x_{0}}(x)\,\mathrm{d}x>CR^{\alpha}\qquad\text{for
all $\alpha\in(0,2)$ and all $C>0$,}$

and

$\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{x_{0}}\|_{L^{p}(\mathbb{R}^{N})}=\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}=\infty\qquad\text{for
all $p>1$.}$

###### Proof.

By Theorem 5.35 in [17],

$\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x)\eqsim|x-x_{0}|^{-N}\big{(}-\log(|x|^{2}))^{-2}\eqsim|x-x_{0}|^{-N}\big{(}\log(|x|))^{-2}\qquad\text{as
$|x|\to 0$.}$

Then for small enough $0<R<<1$,

$\int_{B_{R}(0)}\mathbb{G}_{-\mathcal{L}^{\mu}}^{0}(x)\,\mathrm{d}x\eqsim\int_{0}^{R}r^{-1}(\log(r))^{-2}\,\mathrm{d}r\eqsim\Big{(}\log\Big{(}\frac{1}{R}\Big{)}\Big{)}^{-1}.$

Now, the statement

$\Big{(}\log\Big{(}\frac{1}{R}\Big{)}\Big{)}^{-1}>CR^{\alpha}\qquad\text{for
some $0<R<<1$}$

is equivalent to

$\frac{1}{R}<\textup{exp}\Big{(}C\Big{(}\frac{1}{R}\Big{)}^{\alpha}\Big{)}\qquad\text{for
some $0<R<<1$,}$

which is clearly true for all $\alpha\in(0,2)$ and all $C>0$.

We already know that

$\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}=\int_{0}^{\infty}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\,\mathrm{d}t.$

By Theorem 5.5.2 in [17], for all $0<t\leq\min\\{1,N/(2\alpha)\\}$,

$\begin{split}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}^{p}&\geq
Ct^{p}\bigg{(}\int_{|x|<1}|x|^{-p(N-t\alpha)}\,\mathrm{d}x+\int_{|x|>1}|x|^{-p(N+\alpha)}\,\mathrm{d}x\bigg{)}\\\
&\gtrsim t^{p}\int_{0}^{1}r^{N-1-p(N-t\alpha)}\,\mathrm{d}r.\end{split}$

If $1<p\leq 2$, then

$\frac{N}{\alpha}\frac{p-1}{p}\leq\min\Big{\\{}1,\frac{N}{2\alpha}\Big{\\}},$

and

$\begin{split}\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}&\geq\int_{0}^{\frac{N}{\alpha}\frac{p-1}{p}}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\,\mathrm{d}t\\\
&\gtrsim\int_{0}^{\frac{N}{\alpha}\frac{p-1}{p}}\textup{e}^{-t}t\bigg{(}t^{p}\lim_{\xi\to
0^{+}}\int_{\xi}^{1}r^{N-1-p(N-t\alpha)}\,\mathrm{d}r\bigg{)}^{\frac{1}{p}}\,\mathrm{d}t\\\
&=\int_{0}^{\frac{N}{\alpha}\frac{p-1}{p}}\textup{e}^{-t}t^{2}\bigg{(}\frac{1}{p\alpha}\frac{1}{\frac{N}{\alpha}\frac{(p-1)}{p}-t}\bigg{)}^{\frac{1}{p}}\bigg{(}\lim_{\xi\to
0^{+}}\frac{1}{\xi^{p\alpha(\frac{N}{\alpha}\frac{p-1}{p}-t)}}-1\bigg{)}^{\frac{1}{p}}\,\mathrm{d}t\\\
&\geq\int_{0}^{\frac{N}{\alpha}\frac{p-1}{p}}\textup{e}^{-t}t^{2}\bigg{(}\frac{1}{N(p-1)}\bigg{)}^{\frac{1}{p}}\bigg{(}\lim_{\xi\to
0^{+}}\frac{1}{\xi^{p\alpha(\frac{N}{\alpha}\frac{p-1}{p}-t)}}-1\bigg{)}^{\frac{1}{p}}\,\mathrm{d}t=\infty.\end{split}$

If $p>2$, then

$\frac{N}{\alpha}\frac{p-1}{p}>\frac{N}{\alpha}\frac{1}{2},$

and we simply consider

$\|\mathbb{G}_{I-\mathcal{L}^{\mu}}^{0}\|_{L^{p}(\mathbb{R}^{N})}\geq\int_{0}^{\min\\{1,N/(2\alpha)\\}}\textup{e}^{-t}\|\mathbb{H}_{-\mathcal{L}^{\mu}}^{0}(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}\,\mathrm{d}t$

to reach the same conclusion. ∎

## Acknowledgements

J. Endal has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement no.
839749 “Novel techniques for quantitative behavior of convection-diffusion
equations (techFRONT)”, and from the Research Council of Norway under the
MSCA-TOPP-UT grant agreement no. 312021.

M. Bonforte was partially supported by the Projects MTM2017-85757-P and
PID2020-113596GB-I00 (Spanish Ministry of Science and Innovation). M. Bonforte
moreover acknowledges financial support from the Spanish Ministry of Science
and Innovation, through the “Severo Ochoa Programme for Centres of Excellence
in R&D” (CEX2019-000904-S) and by the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement no.
777822.

We would like to thank Nikita Simonov for fruitful discussions, and the
anonymous referee for a thorough reading and insightful suggestions that
helped us improve the paper.

Conflict of interest statement. On behalf of all authors, the corresponding
author states that there is no conflict of interest.

Data Availability Statements. All data generated or analysed during this study
are included in this published article.

##  Appendix A Technical lemmas

Implicitly, we use the following in the Moser iteration:

######  Lemma A.1.

Assume $K(p)>0$ is such that $\lim_{p\to\infty}K(p)<\infty$, $p_{0}\geq 1$,
and

$\psi\in
L^{p}(\mathbb{R}^{N})\qquad\text{and}\qquad\|\psi\|_{L^{p}(\mathbb{R}^{N})}\leq
K(p)\quad\text{for all $p\in[p_{0},\infty)$.}$

Then $\psi\in L^{\infty}(\mathbb{R}^{N})$, and moreover,

$\|\psi\|_{L^{\infty}(\mathbb{R}^{N})}\leq\lim_{p\to\infty}K(p).$

###### Proof.

Define $K(\infty):=\lim_{p\to\infty}K(p)$, and consider

$\Psi:=|\psi|\mathbf{1}_{\\{|\psi|\leq
K(\infty)+1\\}}+(K(\infty)+1)\mathbf{1}_{\\{|\psi|>K(\infty)+1\\}}=\min\\{|\psi|,K(\infty)+1\\}$

from which it follows that $\Psi\leq K(\infty)+1$ and $\Psi\leq|\psi|$. Then
$\Psi\in L^{\infty}(\mathbb{R}^{N})$ and
$\|\Psi\|_{L^{p}}\leq\|\psi\|_{L^{p}}\leq K(p)$ for all $p\in[p_{0},\infty)$,
and hence, $\|\Psi\|_{L^{\infty}}=\lim_{p\to\infty}\|\Psi\|_{L^{p}}\leq
K(\infty)$. But then $\min\\{|\psi|,K(\infty)+1\\}\leq K(\infty)$ which
implies $\|\psi\|_{L^{\infty}}\leq K(\infty)$. ∎

The next lemma is classical, but we state and prove it for completeness.

######  Lemma A.2 (A DeGiorgi-type lemma).

Assume that $z\in\mathbb{R}^{N}$, and that $f\in L^{\infty}(B_{3R}(z))$ with
$R>0$ fixed. If, for any $R\leq\rho<\bar{\rho}\leq 3R$, some $\delta\in(0,1)$,
and some $M>0$ independent of $\rho,\bar{\rho}$, we have that

$\|f\|_{L^{\infty}(B_{\rho}(z))}\leq\delta\|f\|_{L^{\infty}(B_{\bar{\rho}}(z))}+M,$

then

$\|f\|_{L^{\infty}(B_{\rho}(z))}\leq\frac{1}{1-\delta}M.$

###### Proof.

We follow the proof of Lemma 1.2 in Chapter 4 of [80]. Fix $\rho\geq R$. For
some $0<\eta<1$ we consider the sequence $\\{\rho_{i}\\}$ defined recursively
by

$\rho_{0}=\rho\qquad\text{and}\qquad\rho_{i+1}:=\rho_{i}+(1-\eta)\eta^{i}\rho.$

Note that $\rho_{\infty}=2\rho$. Since
$2\rho=\rho_{\infty}>\ldots>\rho_{1}>\rho_{0}=\rho$,

$\begin{split}\|f\|_{L^{\infty}(B_{\rho}(z))}&=\|f\|_{L^{\infty}(B_{\rho_{0}}(z))}\leq\delta\|f\|_{L^{\infty}(B_{\rho_{1}}(z))}+M\leq\delta^{2}\|f\|_{L^{\infty}(B_{\rho_{2}}(z))}+(1+\delta)M\\\
&\leq\ldots\leq\delta^{k}\|f\|_{L^{\infty}(B_{\rho_{k}}(z))}+M\sum_{i=0}^{k-1}\delta^{i}.\end{split}$

The conclusion follows by letting $k\to\infty$. ∎

##  Appendix B $L^{1}$–$L^{\infty}$-smoothing controls
$L^{q}$–$L^{p}$-smoothing

Throughout this section, $F>0$ is some nonincreasing function and $C>0$ is
some constant (which might change from line to line) not depending on any norm
of $u$.

######  Theorem B.1.

Assume that $0\leq u_{0}=u(\cdot,0)\in(L^{1}\cap L^{\infty})(\mathbb{R}^{N})$,
and that, for a.e. $t_{1}>t_{0}\geq 0$,

$\|u(t_{1})\|_{L^{\infty}}\leq
F(t_{1}-t_{0})\|u(t_{0})\|_{L^{q}}^{\gamma}\qquad\text{for some
$0\leq\gamma<1$ and $q\in[1,\infty)$.}$

Then, for all $1\leq q\leq p\leq\infty$,

$\|u(t_{1})\|_{L^{p}}\leq G(t_{1}-t_{0})H(\|u(t_{0})\|_{L^{q}}),$

where $G,H\geq 0$ are functions depending on $F,\gamma,p,q$ and $\gamma,p,q$,
respectively, which has to be determined in each case.

The proof is a consequence of several results in this section. We start by
investigating immediate consequences of $L^{q}$–$L^{\infty}$-smoothing effects
through Young and Hölder inequalities, in addition to a DeGiorgi type lemma.

######  Lemma B.2.

Assume that $0\leq u_{0}=u(\cdot,0)\in(L^{1}\cap L^{\infty})(\mathbb{R}^{N})$,
and that, for a.e. $t_{1}>t_{0}\geq 0$,

$\|u(t_{1})\|_{L^{\infty}}\leq
F(t_{1}-t_{0})\|u(t_{0})\|_{L^{q}}^{\gamma}\qquad\text{for some
$0\leq\gamma<1$ and $q\in[1,\infty)$.}$

Then:

  1. (a)

($L^{\leq q}$–$L^{\infty}$-smoothing)

$\|u(t_{1})\|_{L^{\infty}}\leq CF(t_{1}-t_{0})^{\frac{q}{(1-\gamma)q+\gamma
r}}\|u(t_{0})\|_{L^{r}}^{\frac{\gamma r}{(1-\gamma)q+\gamma r}}\qquad\text{for
$r\in[1,q]$.}$

  2. (b)

($L^{\leq q}$–$L^{>q}$-smoothing)

$\|u(t_{1})\|_{L^{p}}\leq
CF(t_{1}-t_{0})^{\frac{q}{p}\frac{p-r}{(1-\gamma)q+\gamma
r}}\|u(t_{0})\|_{L^{r}}^{\frac{r}{p}\frac{(1-\gamma)q+\gamma
p}{(1-\gamma)q+\gamma r}}\qquad\text{for $p\in(r,\infty)$ and $r\in[1,q]$.}$

######  Remark B.3.

The homogeneous smoothing estimates (cf. Theorem 3.1(a) and Theorem 3.7) can
respectively be recovered by choosing $q=1=r$ and:

  1. (i)

If $\gamma=\alpha\theta$, then

$\frac{\gamma(p-1)+1}{p}=\frac{1}{p}\frac{\theta_{1}}{\theta_{p}}.$

  2. (ii)

If $\gamma=0$, then

$\frac{\gamma(p-1)+1}{p}=\frac{1}{p}.$

###### Proof of Lemma B.2.

(a) Since $q\geq r\geq 1$, we have

$\|u(t_{0})\|_{L^{q}}^{\gamma}\leq\|u(t_{0})\|_{L^{\infty}}^{\frac{\gamma(q-r)}{q}}\|u(t_{0})\|_{L^{r}}^{\frac{\gamma
r}{q}}.$

Applying the Young inequality (1.7) with $\vartheta=\frac{q}{\gamma(q-r)}>1$
yields

$\|u(t_{1})\|_{L^{\infty}}\leq\frac{\gamma(q-r)}{q}\|u(t_{0})\|_{L^{\infty}}+CF(t_{1}-t_{0})^{\frac{q}{(1-\gamma)q+\gamma
r}}\|u(t_{0})\|_{L^{r}}^{\frac{\gamma r}{(1-\gamma)q+\gamma r}}.$

Since $\gamma(q-r)/q<1$, we can reabsorb the $L^{\infty}$-norm by a variant of
a classical lemma due to DeGiorgi (cf. Lemma A.2).

(b) When $p\geq r\geq 1$, we use Proposition 4.1(b)(ii) to get

$\|u(t_{1})\|_{L^{p}}\leq\|u(t_{1})\|_{L^{\infty}}^{\frac{p-r}{p}}\|u(t_{1})\|_{L^{r}}^{\frac{r}{p}}\leq\|u(t_{1})\|_{L^{\infty}}^{\frac{p-r}{p}}\|u(t_{0})\|_{L^{r}}^{\frac{r}{p}}.$

Now, part (a) yields

$\begin{split}\|u(t_{1})\|_{L^{p}}&\leq
CF(t_{1}-t_{0})^{\frac{q}{(1-\gamma)q+\gamma
r}\frac{p-r}{p}}\|u(t_{0})\|_{L^{r}}^{\frac{\gamma r}{(1-\gamma)q+\gamma
r}\frac{p-r}{p}}\|u(t_{0})\|_{L^{r}}^{\frac{r}{p}}\\\
&=CF(t_{1}-t_{0})^{\frac{q}{p}\frac{p-r}{(1-\gamma)q+\gamma
r}}\|u(t_{0})\|_{L^{r}}^{\frac{r}{p}\frac{(1-\gamma)q+\gamma
p}{(1-\gamma)q+\gamma r}}.\end{split}$

The proof is complete. ∎

As we saw, the easy consequences of smoothing effects, is to loose
integrability on the right-hand side. Now, we instead want to gain
integrability, i.e., we want $L^{1}$–$L^{\infty}$ to $L^{\geq
1}$–$L^{\infty}$.

To gain integrability, however, requires a refined technique. In bounded
domains $|\Omega|<\infty$, this is usually accomplished by the fact that
$L^{\tilde{q}}\subseteq L^{q}$ with $1\leq q\leq\tilde{q}\leq\infty$, i.e., by
the Hölder inequality

$\|u\|_{L^{q}(\Omega)}^{q}=\int_{\Omega}|u|^{q}\,\mathrm{d}x\leq\bigg{(}\int_{\Omega}\big{(}|u|^{q}\big{)}^{\frac{\tilde{q}}{q}}\,\mathrm{d}x\bigg{)}^{\frac{q}{\tilde{q}}}\bigg{(}\int_{\Omega}\big{(}1\big{)}^{\frac{\tilde{q}}{\tilde{q}-q}}\,\mathrm{d}x\bigg{)}^{\frac{\tilde{q}-q}{\tilde{q}}}=\|u\|_{L^{\tilde{q}}(\Omega)}^{q}|\Omega|^{\frac{\tilde{q}-q}{\tilde{q}}}.$

So, while such a statement is trivial in bounded domains, the story is quite
different in $\mathbb{R}^{N}$. The reason for this can be seen by the
following estimate:

$\|u\|_{L^{q}}^{q}=\int_{B_{R}(0)}|u|^{q}+\int_{\mathbb{R}^{N}\setminus
B_{R}(0)}|u|^{q}.$

On the small ball, we use that $L^{\tilde{q}}\subseteq L^{q}$, but what to do
on the complementary set? In fact, there the “natural” ordering is

$\frac{1}{(1+|x|)^{\tilde{q}}}\leq\frac{1}{(1+|x|)^{q}},$

i.e., opposite of the small ball. Hence,

$\int_{\mathbb{R}^{N}\setminus B_{R}(0)}|u|^{q}\leq
C_{R}\bigg{(}\int_{\mathbb{R}^{N}\setminus
B_{R}(0)}|u|^{\tilde{q}}\bigg{)}^{\frac{q}{\tilde{q}}}$

cannot be true for all functions, and must be a property of the equation
itself.

We therefore rely on a nice idea taken from Section 3.1 in [113]: Consider
(GPME) with the nonlinearity

$\varphi:r\mapsto(r+\varepsilon)^{m}-\varepsilon^{m}\qquad\text{for some
$\varepsilon>0$ and some $m>1$.}$

In the case of Lévy operators (1.1) with $c=0$, this equation has been studied
in [63], and in the case of the more general setting of possibly $x$-dependent
$\mathfrak{m}$-accretive operators, this equation has been studied in detail
in e.g. [22, Appendix B] and [57]. E.g., existence, uniqueness, and the
comparison principle holds for sign-changing solutions. In our setting, we
also have:

######  Lemma B.4.

Assume ($\textup{H}_{m}$) and $\varepsilon>0$. Let $u$ be a weak dual solution
of (GPME) with initial data $0\leq u_{0}\in L^{1}(\mathbb{R}^{N})$, and $v$ be
a weak dual solution of (GPME) with nonlinearity $\varphi$ and initial data
$0\leq v_{0}\in L^{1}(\mathbb{R}^{N})$. Under suitable assumptions on the
associated Green function:

(i) For a.e. $t_{1}>t_{0}\geq 0$,

$\|u(t_{1})\|_{L^{\infty}}\leq
F(t_{1}-t_{0})\|u(t_{0})\|_{L^{1}}^{\gamma}\qquad\text{for some
$0\leq\gamma<1$.}$

If, moreover, $u(x,t)\leq v(x,t)+\varepsilon$ for a.e. $(x,t)\in Q_{T}$, then:

(ii) For a.e. $t_{1}>t_{0}\geq 0$,

$\|v(t_{1})\|_{L^{\infty}}\leq
F(t_{1}-t_{0})\|v(t_{0})\|_{L^{1}}^{\gamma}+C\varepsilon,$

with the same $F,\gamma$ as given above.

###### Proof.

(i) This already holds, see Theorems 3.1 and 3.7.

(ii) Even though the nonlinearity $\varphi(r)$ is different from $r^{m}$, we
will see that we can repeat the steps leading up to Theorems 3.1 and 3.7 when
we in addition know that $u\leq v+\varepsilon$.

Recall that by Lemma 4.4, we have

$t\mapsto t^{\frac{m}{m-1}}u^{m}(\cdot,t)\qquad\text{is nondecreasing for a.e.
$x\in\mathbb{R}^{N}$,}$

and since $u\leq v+\varepsilon$, we get the following time-monotonicity for
$v+\varepsilon$:

$t\mapsto t^{\frac{m}{m-1}}(v(\cdot,t)+\varepsilon)^{m}\qquad\text{is
nondecreasing for a.e. $x\in\mathbb{R}^{N}$.}$

Then

$\begin{split}&\int_{\tau_{*}}^{\tau}(v(x_{0},t)+\varepsilon)^{m}\,\mathrm{d}t-\varepsilon^{m}(\tau-\tau_{*})=\int_{\tau_{*}}^{\tau}\varphi(v(x_{0},t))\,\mathrm{d}t\\\
&=\int_{\mathbb{R}^{N}}v(x,\tau_{*})\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x-\int_{\mathbb{R}^{N}}v(x,\tau)\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x\leq\int_{\mathbb{R}^{N}}v(x,\tau_{*})\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x,\end{split}$

or

$\begin{split}\int_{\tau_{*}}^{\tau}(v(x_{0},t)+\varepsilon)^{m}\,\mathrm{d}t\leq\int_{\mathbb{R}^{N}}v(x,\tau_{*})\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x+\varepsilon^{m}(\tau-\tau_{*}).\end{split}$

By the time-monotonicity,

$(v(x_{0},\tau_{*})+\varepsilon)^{m}\leq\frac{C(m)}{\tau_{*}}\int_{\mathbb{R}^{N}}v(x,\tau_{*})\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x+C(m)\varepsilon^{m}.$

Since $r\mapsto r^{m}$ is superadditive on $\mathbb{R}_{+}$, we also get

$v^{m}(x_{0},\tau_{*})\leq\frac{C(m)}{\tau_{*}}\int_{\mathbb{R}^{N}}v(x,\tau_{*})\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\,\mathrm{d}x+(C(m)-1)\varepsilon^{m}.$

Hence, we get the stated $L^{1}$–$L^{\infty}$-smoothing for $v$ by simply
following the proof for $u$, and using that $r\mapsto r^{\frac{1}{m}}$ is
subadditive on $\mathbb{R}_{+}$. ∎

######  Proposition B.5.

Assume that $0\leq u_{0}=u(\cdot,0)\in(L^{1}\cap L^{p})(\mathbb{R}^{N})$ for
some $p\in(1,\infty)$, and that, for a.e. $t_{1}>t_{0}\geq 0$,

$\|u(t_{1})\|_{L^{\infty}}\leq
F(t_{1}-t_{0})\|u(t_{0})\|_{L^{1}}^{\gamma}\qquad\text{for some
$\gamma\in[0,1)$.}$

Moreover, if the comparison principle holds for sign-changing weak dual
solutions of (GPME) with the nonlinearity $\varphi$, then

$\|u(t_{1})\|_{L^{\infty}}\leq
CF(t_{1}-t_{0})^{\frac{1}{\gamma(p-1)+1}}\|u(t_{0})\|_{L^{p}}^{\frac{\gamma
p}{\gamma(p-1)+1}}.$

######  Remark B.6.

The comparison principle indeed holds for sign-changing weak dual solutions of
(GPME) with the nonlinearity $\varphi$: We simply repeat the existence proof
in this setting.

######  Remark B.7.

Let us check that we indeed recover the different homogeneous cases:

  1. (i)

If $\gamma_{1}=\gamma=\alpha\theta$, $F(t)=Ct^{-\gamma_{2}}$, and
$\gamma_{2}=N\theta$, then

$\frac{\gamma_{1}p}{\gamma_{1}(p-1)+1}=\alpha
p\theta_{p}\qquad\text{and}\qquad\frac{\gamma_{2}}{\gamma_{1}(p-1)+1}=N\theta_{p}.$

  2. (ii)

If $\gamma_{1}=\gamma=0$, $F(t)=Ct^{-\gamma_{2}}$, and $\gamma_{2}=1/(m-1)$,
then

$\frac{\gamma_{1}p}{\gamma_{1}(p-1)+1}=0\qquad\text{and}\qquad\frac{\gamma_{2}}{\gamma_{1}(p-1)+1}=\frac{1}{m-1}.$

###### Proof of Proposition B.5.

Consider the function $v_{\varepsilon}:=u-\varepsilon$ with $\varepsilon>0$,
where $u$ solves (GPME) with initial data $u_{0}\geq 0$. Note that
$v_{\varepsilon}$ also solves (GPME) with $\varphi$ as nonlinearity (we have
subtracted the term $\varepsilon^{m}$ for normalization purposes), and
unsigned initial data $u_{0}-\varepsilon$.

Now, consider the solution $\tilde{v}_{\varepsilon}$ of (GPME) with
nonlinearity $\varphi$ and initial data $(u_{0}-\varepsilon)_{+}$. Due to the
Hölder inequality,

$\begin{split}\int(u_{0}-\varepsilon)_{+}=\int_{\\{u_{0}>\varepsilon\\}}(u_{0}-\varepsilon)\leq\int_{\\{u_{0}>\varepsilon\\}}u_{0}=\int
u_{0}\mathbf{1}_{\\{u_{0}>\varepsilon\\}}\leq\|u_{0}\|_{L^{p}}|\\{u_{0}>\varepsilon\\}|^{\frac{p-1}{p}}.\end{split}$

Moreover, for any $f\geq 0$,

$\|f\|_{L^{p}}^{p}=\int_{\\{0\leq
f\leq\varepsilon\\}}|f|^{p}+\int_{\\{f>\varepsilon\\}}|f|^{p}\geq\int_{\\{f>\varepsilon\\}}f^{p}\geq\varepsilon^{p}|\\{f>\varepsilon\\}|.$

Hence,

$\int(u_{0}-\varepsilon)_{+}\leq\frac{\|u_{0}\|_{L^{p}}^{p}}{\varepsilon^{p-1}}.$

In particular, $(u_{0}-\varepsilon)_{+}\in L^{1}$ as long as $u_{0}\in L^{p}$.

The comparison principle for sign-changing solutions of (GPME) with
nonlinearity $\varphi$ then gives

$u_{0}(x)-\varepsilon\leq(u_{0}(x)-\varepsilon)_{+}\qquad\implies\qquad
v_{\varepsilon}(x,t)\leq\tilde{v}_{\varepsilon}(x,t).$

From which we conclude that

$u(x,t)\leq\tilde{v}_{\varepsilon}(x,t)+\varepsilon\qquad\implies\qquad\|u(t)\|_{L^{\infty}}\leq\|\tilde{v}_{\varepsilon}(t)\|_{L^{\infty}}+\varepsilon.$

We are then in the setting of Lemma B.4, and, for all $\varepsilon>0$,

$\|u(t)\|_{L^{\infty}}\leq\|\tilde{v}_{\varepsilon}(t)\|_{L^{\infty}}+\varepsilon\leq
F(t)\|(u_{0}-\varepsilon)_{+}\|_{L^{1}}^{\gamma}+c\varepsilon\leq
F(t)\|u_{0}\|_{L^{p}}^{\gamma p}\varepsilon^{-\gamma(p-1)}+c\varepsilon.$

To conclude, we infimize over $\varepsilon>0$. ∎

##  Appendix C Densely defined, $\mathfrak{m}$-accretive, and Dirichlet in
$L^{1}(\mathbb{R}^{N})$

The existence proof for weak dual solutions is based on the concept of _mild_
solutions, which again relies on finding a.e.-solutions (!) of the
corresponding elliptic problem

$\forall\lambda>0\qquad u+\lambda A[u^{m}]=f\qquad\text{in $\mathbb{R}^{N}$.}$

We will therefore study so-called $\mathfrak{m}$-accretive operators $A$.

###  C.1. The setting of abstract solutions

The Laplacian $(-\Delta)$ (as well as $r\mapsto(-\Delta)[r^{m}]$) is a well-
known example of an operator which is $\mathfrak{m}$_-accretive_ in
$L^{1}(\mathbb{R}^{N})$ [114, Section 10.3.2]. Now, we want to establish that
any _symmetric, nonlocal_ and _constant coefficient_ Lévy operator

$(-\mathcal{L}^{\mu})[\psi]=-P.V.\int_{\mathbb{R}^{N}\setminus\\{0\\}}\big{(}\psi(x+z)-\psi(x)\big{)}\,\mathrm{d}\mu(z)$

is also $\mathfrak{m}$-accretive in $L^{1}(\mathbb{R}^{N})$. Since that
operator is moreover _Dirichlet_ , we get by Propositions 1 and 2 in [52],
that $r\mapsto(-\mathcal{L}^{\mu})[r^{m}]$ is also $\mathfrak{m}$-accretive
and Dirichlet in $L^{1}(\mathbb{R}^{N})$. Indeed, such a result should be
well-known, but we did not manage to find a useful reference for it.

Throughout this section, we stick to the usual notation
$A:=(-\mathcal{L}^{\mu})$.

######  Theorem C.1.

Assume (1). Then the linear operator $A:D(A)\subset L^{1}(\mathbb{R}^{N})\to
L^{1}(\mathbb{R}^{N})$ satisfies:

  1. (i)

$\overline{D(A)}^{\|\cdot\|_{L^{1}(\mathbb{R}^{N})}}=L^{1}(\mathbb{R}^{N})$.

  2. (ii)

$A$ is accretive in $L^{1}(\mathbb{R}^{N})$.

  3. (iii)

$R(I+\lambda A)=L^{1}(\mathbb{R}^{N})$ for all $\lambda>0$.

  4. (iv)

If $f\in L^{1}(\mathbb{R}^{N})$ and $a,b\in\mathbb{R}$ such that $a\leq f\leq
b$ a.e., then $a\leq(I+\lambda A)^{-1}f\leq b$ a.e.

That is, the linear operator $A$ is densely defined, $\mathfrak{m}$-accretive,
and Dirichlet in $L^{1}(\mathbb{R}^{N})$.

######  Remark C.2.

  1. (a)

Since $D(A)\subset L^{1}(\mathbb{R}^{N})$, we define

$D(A):=\\{\psi\in L^{1}(\mathbb{R}^{N})\,:\,\widetilde{A}[\psi]\in
L^{1}(\mathbb{R}^{N})\\}$

where $\widetilde{A}$ is the extension of $A$ to $L^{1}(\mathbb{R}^{N})$ (see
(C.2) below). Moreover, in our case it is well-known that

$A:C_{\textup{c}}^{\infty}(\mathbb{R}^{N})\subset L^{1}(\mathbb{R}^{N})\to
L^{p}(\mathbb{R}^{N})\qquad\text{for all $p\in[1,\infty]$}.$

Hence, $C_{\textup{c}}^{\infty}(\mathbb{R}^{N})\subset D(A)$, and thus, we
have already proven Theorem C.1(i). However, we need to make sure that when we
define the extension $\widetilde{A}$ (see (C.2) below), we have

$\widetilde{A}|_{C_{\textup{c}}^{\infty}(\mathbb{R}^{N})}=A\qquad\text{a.e. in
$\mathbb{R}^{N}$.}$

  2. (b)

In our $\mathbb{R}^{N}$-case, the numbers $a,b$ in Theorem C.1(iv) has the
natural restriction $a\leq b$, $a\leq 0$, and $b\geq 0$.

######  Corollary C.3 ([52, Propositions 1 and 2]).

Assume (1) and ($\textup{H}_{m}$). Then the nonlinear operator $r\mapsto
Ar^{m}:D(Ar^{m})\subset L^{1}(\mathbb{R}^{N})\to L^{1}(\mathbb{R}^{N})$ is
densely defined, $\mathfrak{m}$-accretive, and Dirichlet in
$L^{1}(\mathbb{R}^{N})$.

In particular, for all $f\in L^{1}(\mathbb{R}^{N})$ such that $a\leq f\leq b$,
there exists a unique $u\in L^{1}(\mathbb{R}^{N})$ which satisfies $a\leq
u\leq b$, the comparison principle, $L^{p}$-decay estimate, and

$u+\lambda\widetilde{A}[u^{m}]=f\qquad\text{a.e. in $\mathbb{R}^{N}$.}$

Let us start by proving the range condition. To do so, consider

(C.1) $\forall\lambda>0\qquad u+\lambda A[u]=f\qquad\text{in
$\mathbb{R}^{N}$}.$

######  Definition C.1 (Very weak solutions).

Assume (1). We say that $u\in L_{\textup{loc}}^{1}(\mathbb{R}^{N})$ is a _very
weak solution_ of (C.1) with right-hand side $f\in
L_{\textup{loc}}^{1}(\mathbb{R}^{N})$ if

$\begin{split}\int_{\mathbb{R}^{N}}u\psi\,\mathrm{d}x+\lambda\int_{\mathbb{R}^{N}}uA[\psi]\,\mathrm{d}x=\int_{\mathbb{R}^{N}}f\psi\,\mathrm{d}x\qquad\text{for
all $\psi\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$.}\end{split}$

We need the following result (take $u\mapsto\lambda u$ for all $\lambda>0$ and
choose $\varepsilon=1/\lambda$):

######  Lemma C.4 ([63, Theorem 3.1]).

Assume (1).

  1. (a)

If $f\in C_{\textup{b}}^{\infty}(\mathbb{R}^{N})$, then there exists a unique
classical solution $u\in C_{\textup{b}}^{\infty}(\mathbb{R}^{N})$ of (C.1).
Moreover, for each multiindex $\alpha\in\mathbb{N}^{N}$,

$\|D^{\alpha}u\|_{L^{\infty}(\mathbb{R}^{N})}\leq\|D^{\alpha}f\|_{L^{\infty}(\mathbb{R}^{N})}.$

  2. (b)

If $f\in L^{\infty}(\mathbb{R}^{N})$, then there exists a unique classical
solution $u\in L^{\infty}(\mathbb{R}^{N})$ of (C.1). Moreover,

$\|u\|_{L^{\infty}(\mathbb{R}^{N})}\leq\|f\|_{L^{\infty}(\mathbb{R}^{N})}.$

  3. (c)

If $f\in L^{1}(\mathbb{R}^{N})$, then there exists a unique classical solution
$u\in L^{1}(\mathbb{R}^{N})$ of (C.1). Moreover,

$\|u\|_{L^{1}(\mathbb{R}^{N})}\leq\|f\|_{L^{1}(\mathbb{R}^{N})}.$

######  Remark C.5.

Theorem 6.15 in [63] gives comparison (or $T$-contraction) as well: If $f\in
L_{\textup{loc}}^{1}(\mathbb{R}^{N})$ such that $(f)^{+}\in
L^{1}(\mathbb{R}^{N})$ and $u\in L_{\textup{loc}}^{1}(\mathbb{R}^{N})$ is a
very weak solution of (C.1), then

$\int_{\mathbb{R}^{N}}(u)^{+}\,\mathrm{d}x\leq\int_{\mathbb{R}^{N}}(f)^{+}\,\mathrm{d}x.$

######  Lemma C.6.

Assume $\eqref{muas}$, $p\in(1,\infty]$, $f\in(L^{1}\cap
L^{\infty})(\mathbb{R}^{N})$. Let $u$ be the very weak solution of (C.1) with
right-hand side $f$. Then

$\lambda\|\widetilde{A}[u]\|_{L^{p}(\mathbb{R}^{N})}\leq
2\|f\|_{L^{p}(\mathbb{R}^{N})},$

where the extension $\widetilde{A}:D(A)\cap L^{p}(\mathbb{R}^{N})\to
L^{p}(\mathbb{R}^{N})$ satisfies

(C.2)
$\int_{\mathbb{R}^{N}}\widetilde{A}[u]\psi\,\mathrm{d}x=\int_{\mathbb{R}^{N}}uA[\psi]\,\mathrm{d}x\qquad\text{for
all $\psi\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$.}$

The proof follows after an immediate consequence.

######  Corollary C.7.

Assume $\eqref{muas}$, $p\in(1,\infty]$, $f\in(L^{1}\cap
L^{\infty})(\mathbb{R}^{N})$. Then the equation (C.1) (with
$A\mapsto\widetilde{A}$) holds a.e. in $\mathbb{R}^{N}$.

###### Proof of Lemma C.6.

Definition C.1 gives

$\begin{split}\lambda\int_{\mathbb{R}^{N}}\widetilde{A}[u]\psi\,\mathrm{d}x=\lambda\int_{\mathbb{R}^{N}}uA[\psi]\,\mathrm{d}x=\int_{\mathbb{R}^{N}}(f-u)\psi\,\mathrm{d}x.\end{split}$

Now, take $q\in[1,\infty)$ such that $p^{-1}+q^{-1}=1$. We recall Theorem 2.14
in [94],

$\|\phi\|_{L^{p}(\mathbb{R}^{N})}=\sup_{\|\psi\|_{L^{q}(\mathbb{R}^{N})}\leq
1}\bigg{|}\int_{\mathbb{R}^{N}}\phi(x)\psi(x)\,\mathrm{d}x\bigg{|},$

to obtain, by the Hölder inequality,

$\begin{split}\lambda\|\widetilde{A}[u]\|_{L^{p}(\mathbb{R}^{N})}&=\lambda\sup_{\|\psi\|_{L^{q}(\mathbb{R}^{N})}\leq
1}\bigg{|}\int_{\mathbb{R}^{N}}\widetilde{A}[u]\psi\,\mathrm{d}x\bigg{|}=\sup_{\|\psi\|_{L^{q}(\mathbb{R}^{N})}\leq
1}\bigg{|}\int_{\mathbb{R}^{N}}(f-u)\psi\,\mathrm{d}x\bigg{|}\\\
&\leq\sup_{\|\psi\|_{L^{q}(\mathbb{R}^{N})}\leq
1}\bigg{\\{}\|f-u\|_{L^{p}(\mathbb{R}^{N})}\|\psi\|_{L^{q}(\mathbb{R}^{N})}\bigg{\\}}\leq\|f-u\|_{L^{p}(\mathbb{R}^{N})}\\\
&\leq 2\|f\|_{L^{p}(\mathbb{R}^{N})}.\end{split}$

The proof is complete. ∎

Another consequence is that we can also extend $\widetilde{A}$ to
$L^{1}(\mathbb{R}^{N})$, and then make sense of the equation in
$L^{1}(\mathbb{R}^{N})$. To do so, we follow [40].

######  Corollary C.8.

Assume $\eqref{muas}$ and $f\in(L^{1}\cap L^{\infty})(\mathbb{R}^{N})$. Then

$\lambda\|\widetilde{A}[u]\|_{L^{1}(\mathbb{R}^{N})}\leq
2\|f\|_{L^{1}(\mathbb{R}^{N})}.$

###### Proof.

Since $f\in(L^{1}\cap L^{\infty})(\mathbb{R}^{N})\subset
L^{1}(\mathbb{R}^{N})$, Lemma C.4(c) yields

$\|u\|_{L^{1}(\mathbb{R}^{N})}\leq\|f\|_{L^{1}(\mathbb{R}^{N})}.$

Moreover, by Corollary C.7, equation (C.1) holds pointwise, i.e.,

$\|f-\lambda\widetilde{A}[u]\|_{L^{1}(\mathbb{R}^{N})}\leq\|f\|_{L^{1}(\mathbb{R}^{N})}.$

The reverse triangle inequality then provides the result. ∎

We are now ready to prove Theorem C.1(iii).

######  Proposition C.9.

Assume (1). For all $f\in L^{1}(\mathbb{R}^{N})$, there exists a very weak
solution $u\in L^{1}(\mathbb{R}^{N})$ of (C.1) such that, for all $\lambda>0$,

$u+\lambda\widetilde{A}[u]=f\qquad\text{a.e. in $\mathbb{R}^{N}$,}$

where $\widetilde{A}$ is the extension to $L^{1}(\mathbb{R}^{N})$ of $A$
defined through the relation (C.2).

###### Proof.

Take $\\{f_{n}\\}_{n\in\mathbb{N}}\subset(L^{1}\cap
L^{\infty})(\mathbb{R}^{N})$ such that $f_{n}\to f$ in $L^{1}(\mathbb{R}^{N})$
as $n\to\infty$. By Lemma C.4(c),

$\|u_{n}-u_{m}\|_{L^{1}(\mathbb{R}^{N})}\leq\|f_{n}-f_{m}\|_{L^{1}(\mathbb{R}^{N})}.$

Hence, $\\{u_{n}\\}_{n\in\mathbb{N}}$ is Cauchy in $L^{1}(\mathbb{R}^{N})$ and
there exists a $u\in L^{1}(\mathbb{R}^{N})$ such that $u_{n}\to u$ in
$L^{1}(\mathbb{R}^{N})$. In a similar way, through Corollary C.8,
$\\{\widetilde{A}[u_{n}]\\}_{n\in\mathbb{N}}$ is Cauchy in
$L^{1}(\mathbb{R}^{N})$ and there exists a $U\in L^{1}(\mathbb{R}^{N})$ such
that $\widetilde{A}[u_{n}]\to U$ in $L^{1}(\mathbb{R}^{N})$. The definition of
$\widetilde{A}$ (C.2) then yields

$\int_{\mathbb{R}^{N}}\widetilde{A}[u_{n}]\psi\,\mathrm{d}x=\int_{\mathbb{R}^{N}}u_{n}A[\psi]\,\mathrm{d}x\qquad\text{for
all $\psi\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$}.$

Moreover, since $\psi,A[\psi]\in L^{\infty}(\mathbb{R}^{N})$, the
$L^{1}$-convergence gives $U=\widetilde{A}[u]$. Finally, we take the
$L^{1}$-limit in Definition C.1 and use that fact that all the terms of the
equation are elements in $L^{1}\subset L_{\textup{loc}}^{1}$. ∎

######  Remark C.10.

In the literature, the property

$u_{n}\to u\quad\text{in
$L^{1}(\mathbb{R}^{N})$}\qquad\implies\qquad\widetilde{A}[u_{n}]\to\widetilde{A}[u]\quad\text{in
$L^{1}(\mathbb{R}^{N})$}$

is referred to as the operator $A$ being _closed_ in $L^{1}(\mathbb{R}^{N})$.
Here it automatically follows by the symmetry of the operator and that
$A:C_{\textup{c}}^{\infty}(\mathbb{R}^{N})\to L^{\infty}(\mathbb{R}^{N})$.

Theorem C.1(ii) is a consequence of the $L^{1}$-contraction obtained in Lemma
C.4(c) and Proposition C.9 since then

(C.3)
$\|(I+\lambda\widetilde{A})[u]\|_{L^{1}(\mathbb{R}^{N})}=\|f\|_{L^{1}(\mathbb{R}^{N})}\geq\|u\|_{L^{1}(\mathbb{R}^{N})}.$

We are then in the setting of the classical result:

######  Proposition C.11 (Hille-Yosida/Lumer-Phillips [82, Theorem 4.1.33]).

A linear operator $(A,D(A))$ on $L^{1}(\mathbb{R}^{N})$ is the generator of a
strongly continuous contraction semigroup $(T_{t})_{t\geq 0}$ on
$L^{1}(\mathbb{R}^{N})$ if and only if $A$ satisfies Theorem C.1(i)–(iii).

We, moreover, have that our operators are _maximal accretive_ , i.e.:

######  Proposition C.12 ([12, Proposition 8.3]).

If $A$ is $\mathfrak{m}$-accretive, then $\widetilde{A}=A$.

######  Remark C.13.

Hence, a posteriori, $L^{1}(\mathbb{R}^{N})\ni u\mapsto A[u]$ can be
identified in a unique way as a limit point in $L^{1}(\mathbb{R}^{N})$. See
also Theorem 4.1.40 in [82].

Our next task is Theorem C.1(iv), which follows by:

######  Proposition C.14.

Assume (1) and $a,b\in\mathbb{R}$ such that $a\leq b$, $a\leq 0$, and $b\geq
0$. For all $f\in L^{1}(\mathbb{R}^{N})$ such that

$a\leq f\leq b\qquad\text{a.e.,}$

the unique very weak solution $u\in L^{1}(\mathbb{R}^{N})$ of (C.1) satisfies

$a\leq u\leq b\qquad\text{a.e.}$

###### Proof.

By Remark C.5, the $T$-contraction holds for $L_{\textup{loc}}^{1}$-very weak
solutions of (C.1). On one hand, $f\leq b$ yields $(f-b)^{+}=0\in
L^{1}(\mathbb{R}^{N})$ and then the $T$-contraction gives $u\leq b$. On the
other hand, $a\leq f$ yields $(a-f)^{+}=0\in L^{1}(\mathbb{R}^{N})$ and then
the $T$-contraction gives $a\leq u$.

Here, we used that $a,b$ are very weak solutions with $a,b$ as right-hand
side, and that bounded very weak solutions are unique. ∎

We have then proven that our operator $A$ satisfies Theorem C.1(i)–(iv), which
is exactly the setting of [52].

Finally, let us recall why the above works for our operator
$A=(-\mathcal{L}^{\mu})$. To deduce the $L^{1}$-contraction—or rather the
$T$-contraction—needed to obtain (C.3), we employed a more fundamental result:

######  Lemma C.15.

Assume (1). For all $u\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$,

$\int_{\mathbb{R}^{N}}\widetilde{A}[u]\operatorname{\textup{sign}}^{+}(u)\,\mathrm{d}x\geq
0.$

######  Remark C.16.

This implies the condition stated as Corollary A.13 in [2] or in Proposition
4.6.12 in [82] (see also [103] for $p=1$) from which it follows that $A$ is
accretive.

###### Proof.

Remark C.2(a) gives $\widetilde{A}=A$. By a convex inequality,

$A[u]\operatorname{\textup{sign}}^{+}(u)\geq A[(u)^{+}]\qquad\textup{a.e. in
$\mathbb{R}^{N}$.}$

Now, multiply each side by a smooth cut-off function $\mathcal{X}_{R}\in
C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$ satisfying $0\leq\mathcal{X}_{R}\leq
1$ and $\mathcal{X}_{R}\to 1$ pointwise as $R\to\infty$, integrate, use
symmetry, and that $A[\mathcal{X}_{R}]\to 0$ pointwise as $R\to\infty$. ∎

######  Corollary C.17.

Assume (1). For all $u\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$,

$\int_{\mathbb{R}^{N}}\widetilde{A}[u]\operatorname{\textup{sign}}^{+}(u-1)\,\mathrm{d}x\geq
0.$

######  Remark C.18.

Operators satisfying the above are called Dirichlet operators, see Definition
4.6.7 in [82] (and also [103] for $p=1$). Moreover, according to Proposition
4.6.9 in [82] Dirichlet operators imply Theorem C.1(iv) for both the semigroup
and the resolvent of the semigroup generated by the operator, i.e., the
semigroup and the resolvent of the semigroup are _sub-Markovian_.

###### Proof.

The result can be found as Corollary 3.2 in [103], which actually provides an
equivalence between the two conditions in our setting. Let us include a proof
for completeness.

Again, Remark C.2(a) gives $\widetilde{A}=A$. Now, take $u-\mathcal{X}_{R}\in
C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$ in Lemma C.15. Since
$\operatorname{\textup{sign}}^{+}(u-1)\leq\operatorname{\textup{sign}}^{+}(u-\mathcal{X}_{R})$,
we have that

$\begin{split}&\int_{\mathbb{R}^{N}}A[u]\operatorname{\textup{sign}}^{+}(u-\mathcal{X}_{R})\,\mathrm{d}x\\\
&\geq\int_{\mathbb{R}^{N}}A[\mathcal{X}_{R}]\operatorname{\textup{sign}}^{+}(u-\mathcal{X}_{R})\,\mathrm{d}x\geq\int_{\mathbb{R}^{N}}A[\mathcal{X}_{R}]\operatorname{\textup{sign}}^{+}(u-1)\,\mathrm{d}x.\end{split}$

Moreover,

$\int_{\mathbb{R}^{N}}\operatorname{\textup{sign}}^{+}(u-1)\,\mathrm{d}x=\int_{\\{u>1\\}}1\,\mathrm{d}x<\int_{\\{u>1\\}}u\,\mathrm{d}x\leq\int_{\mathbb{R}^{N}}|u|\,\mathrm{d}x,$

which means that we can, again, use that $A[\mathcal{X}_{R}]\to 0$ pointwise
as $R\to\infty$ on the right-hand side. While on the left-hand side we simply
use the Lebesgue dominated convergence theorem. ∎

######  Remark C.19.

Lemma C.4 is also true for the general operator (possibly local, nonlocal, or
a combination) $\mathfrak{L}^{\sigma,\mu}$ defined by (1.1) with $c=0$, see
[62]. Moreover, if $u$ solves

$\varepsilon u+(-\mathfrak{L}^{\sigma,\mu})[u]=f\qquad\text{in
$\mathbb{R}^{N}$ for all $\varepsilon>0$,}$

then $u$ solves

$(\varepsilon-1)u+(I-\mathfrak{L}^{\sigma,\mu})[u]=f\qquad\text{in
$\mathbb{R}^{N}$ for all $\varepsilon>1$.}$

Now, take $u\mapsto\lambda u$ and choose $\varepsilon=1+1/\lambda$ to obtain
that $u$ solves

$u+\lambda(I-\mathfrak{L}^{\sigma,\mu})[u]=f\qquad\text{in $\mathbb{R}^{N}$
for all $\lambda>0$.}$

Hence, $(-\mathfrak{L}^{\sigma,\mu})$ and $(I-\mathfrak{L}^{\sigma,\mu})$ are
also $\mathfrak{m}$-accretive. The latter is exactly (1.1) with $c=1$.

To prove the Dirichlet property, we used that $A[\textup{const}]=0$ in
Corollary C.14. This is of course true for $(-\mathfrak{L}^{\sigma,\mu})$,
while for $(I-\mathfrak{L}^{\sigma,\mu})$, we have that $f=\textup{const}$
gives $u=\textup{const}/(1+\lambda)$ as a solution. Arguing as before,
however, we still have $f\geq 0$ (resp. $\leq 0$) implies $u\geq 0$ (resp.
$\leq 0$). According to Remark C.2, it remains to check $b>0,a<0$ and $a\leq
f\leq b$: We readily get $a/(1+\lambda)\leq u\leq b/(1+\lambda)$, i.e., $a\leq
u\leq b$ since $\lambda>0$.

We then conclude that both $(-\mathfrak{L}^{\sigma,\mu})$ and
$(I-\mathfrak{L}^{\sigma,\mu})$ satisfy Theorem C.1, Corollary C.3, and
Proposition C.12. So, indeed the whole class of _symmetric_ Lévy operators
with _constant coefficients_ are within the above framework.

###  C.2. The setting of very weak solutions

We will now provide an a priori different approach to the one developed in the
previous subsection.

Very weak solutions of

(C.4) $\forall\lambda>0\qquad u+\lambda A[u^{m}]=f\qquad\text{in
$\mathbb{R}^{N}$}$

can be given as:

######  Definition C.2 (Very weak solutions).

Assume (1). We say that $u\in L_{\textup{loc}}^{1}(\mathbb{R}^{N})$ is a _very
weak solution_ of (C.4) with right-hand side $f\in
L_{\textup{loc}}^{1}(\mathbb{R}^{N})$ if $u^{m}\in
L_{\textup{loc}}^{1}(\mathbb{R}^{N})$ and

$\begin{split}\int_{\mathbb{R}^{N}}u\psi\,\mathrm{d}x+\lambda\int_{\mathbb{R}^{N}}u^{m}A[\psi]\,\mathrm{d}x=\int_{\mathbb{R}^{N}}f\psi\,\mathrm{d}x\qquad\text{for
all $\psi\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$.}\end{split}$

We collect uniqueness from [62, Theorem 3.2], and a priori estimates from [65,
Remark 5.10].

######  Theorem C.20.

Assume $0\leq f\in(L^{1}\cap L^{\infty})(\mathbb{R}^{N})$, ($\textup{H}_{m}$),
(1), and $A=(-\mathfrak{L}^{\sigma,\mu})$.

  1. (a)

There exists a unique very weak solution $0\leq u\in(L^{1}\cap
L^{\infty})(\mathbb{R}^{N})$ of (C.4) with right-hand side $f$.

  2. (b)

Let $u,v$ be two very weak solutions of (C.4) with respective right-hand sides
$f,g$. Then:

    1. (i)

(Comparison) If $f\leq g$, then $u\leq v$.

    2. (ii)

($L^{p}$-decay)
$\|u\|_{L^{p}(\mathbb{R}^{N})}\leq\|f\|_{L^{p}(\mathbb{R}^{N})}$ for all
$p\in[1,\infty]$.

###  C.3. Comparison between abstract and very weak solutions

If $f\in L^{1}(\mathbb{R}^{N})$ only, it is hard to see how to construct very
weak solutions of (C.4), but as in the abstract setting we could require that
$a\leq f\leq b$ for $a,b\in\mathbb{R}$. We in any case have:

######  Lemma C.21.

Assume $0\leq f\in(L^{1}\cap L^{\infty})(\mathbb{R}^{N})$, ($\textup{H}_{m}$),
and (1). If the operator $A$ has an extension $\widetilde{A}$ to
$L^{1}(\mathbb{R}^{N})$, then very weak and a.e.-solutions of (C.4) coincide.

######  Remark C.22.

Here we discover the advantage of $\mathfrak{m}$-accretive operators: By
Proposition C.12, $\widetilde{A}=A$, and thus we obtain an a.e.-equation
involving the operator itself! In addition, the abstract setting does not see
the difference between $A=(-\mathfrak{L}^{\sigma,\mu})$ and
$A=(I-\mathfrak{L}^{\sigma,\mu})$ since they are both
$\mathfrak{m}$-accretive.

###### Proof.

By Corollary C.3, we have a.e.-solutions of (C.4) (with
$A\mapsto\widetilde{A}$). Multiplying by $\psi\in
C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$, integrating over $\mathbb{R}^{N}$,
and using the definition of the extension of the operator (C.2), shows that
those a.e.-solutions are actually very weak solutions in the sense of
Definition C.2. However, we can also start with very weak solutions by Theorem
C.20, use the $L^{1}$-extension of the operator (C.2), and see that the
equation (with $A\mapsto\widetilde{A}$) actually holds a.e. Hence, the
equivalence between the elliptic problems is settled. ∎

##  Appendix D The inverse of a densely defined, $\mathfrak{m}$-accretive,
Dirichlet operator

This section is devoted to showing that a densely defined,
$\mathfrak{m}$-accretive, Dirichlet operator in $L^{1}(\mathbb{R}^{N})$ has an
inverse such that (2) holds, under (possibly) some additional assumptions on
the heat kernel associated with the operator.

Consider a strongly continuous contraction semigroup $(T_{t})_{t\geq 0}$ in
$L^{1}(\mathbb{R}^{N})$ which is moreover sub-Markovian. The discussion before
Definition 3.5.17 in [83] gives that the resolvent $(R_{\lambda})_{\lambda>0}$
of the semigroup is well-defined for functions in $L^{1}(\mathbb{R}^{N})$,
i.e.,

$R_{\lambda}[\psi]:=\int_{0}^{\infty}\textup{e}^{-\lambda
t}T_{t}[\psi]\,\mathrm{d}t<\infty\qquad\text{for all $\psi\in
L^{1}(\mathbb{R}^{N})$.}$

Moreover, the Green (or potential) operator $G$ associated with
$(T_{t})_{t\geq 0}$ is defined as

$G[\psi]:=\lim_{\lambda\to 0^{+}}R_{\lambda}\psi\qquad\text{for all
$0\leq\psi\in L^{1}(\mathbb{R}^{N})$.}$

######  Definition D.1 (Transient).

A strongly continuous sub-Markovian contraction semigroup $(T_{t})_{t\geq 0}$
in $L^{1}(\mathbb{R}^{N})$ which is moreover sub-Markovian is called
_transient_ if

$G[\psi](x)=\int_{0}^{\infty}T_{t}[\psi]\,\mathrm{d}t<\infty\qquad\text{for
all $0\leq\psi\in L^{1}(\mathbb{R}^{N})$.}$

By Proposition C.11, Theorem C.1, and Remark C.19, the operators
$(-\mathfrak{L}^{\sigma,\mu})$ and $(I-\mathfrak{L}^{\sigma,\mu})$ generate
the respective strongly continuous contraction semigroups
$(T_{t}^{-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$ and
$(T_{t}^{I-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$ in $L^{1}(\mathbb{R}^{N})$
which are moreover sub-Markovian. Note that by uniqueness of strongly
continuous semigroups (cf. Corollary 4.1.35 in [82]),
$(T_{t}^{I-\mathfrak{L}^{\sigma,\mu}})_{t\geq
0}=(\textup{e}^{-t}T_{t}^{-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$. An
immediate consequence of Example 3.5.30 in [83] is:

######  Lemma D.1.

The semigroup $(\textup{e}^{-t}T_{t}^{-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$
associated with $(I-\mathfrak{L}^{\sigma,\mu})$, which satisfies Proposition
C.11 and Theorem C.1, is transient.

For the operator $(-\mathfrak{L}^{\sigma,\mu})$, we note that the semigroup
defined as

$T_{t}[\psi]:=\int_{\mathbb{R}^{N}}\psi(x-y)\mathbb{H}_{t}(y)\,\mathrm{d}y\qquad\textup{for
all $\psi\in L^{1}(\mathbb{R}^{N})$}$

is a strongly continuous sub-Markovian contraction semigroup in
$L^{1}(\mathbb{R}^{N})$ with $(-\mathfrak{L}^{\sigma,\mu})$ as generator. This
can easily be seen since $(-\mathfrak{L}^{\sigma,\mu})$ admits a symmetric and
positive heat kernel satisfying $\mathbb{H}_{t}\in L^{1}(\mathbb{R}^{N})$ due
to the fact that the corresponding heat equation enjoys mass conservation,
$L^{1}$-decay, the comparison principle, and has solutions in
$C([0,T];L^{1}(\mathbb{R}^{N}))$. By Corollary 4.1.35 in [82], again, this
semigroup must coincide with $(T_{t}^{-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$.
Moreover:

######  Lemma D.2 ([83, Theorem 3.5.51]).

The semigroup $(T_{t}^{-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$ associated with
$(-\mathfrak{L}^{\sigma,\mu})$, which satisfies Proposition C.11 and Theorem
C.1, is transient if and only if, for all compact $K\subset\mathbb{R}^{N}$,

$\int_{0}^{\infty}\int_{K}\mathbb{H}_{t}(x)\,\mathrm{d}x\,\mathrm{d}t<\infty.$

At this point $G$ is a good candidate for $A^{-1}$, but we need additional
properties like e.g. $G:L^{p}\to D(A)$. The rigorous answer can be found in
Proposition 3.5.79 in [83]. We simply check that

$\lim_{t\to\infty}T_{t}[\psi]\to 0\qquad\text{for all $\psi\in
L^{1}(\mathbb{R}^{N})$.}$

For convolution semigroups with a symmetric positive kernel $\mathbb{H}_{t}$,
we get

$|T_{t}[\psi](x)|\leq\int_{\mathbb{R}^{N}}|\psi(x-y)|\mathbb{H}_{t}(y)\,\mathrm{d}y,$

and hence, this is really a condition on the kernel by the Lebesgue dominated
convergence theorem. Note that we can immediately conclude for
$I-\mathfrak{L}^{\sigma,\mu}$ since
$T_{t}^{I-\mathfrak{L}^{\sigma,\mu}}[\psi]=\textup{e}^{-t}T_{t}^{-\mathfrak{L}^{\sigma,\mu}}[\psi]$,
$\|T_{t}^{-\mathfrak{L}^{\sigma,\mu}}[\psi]\|_{L^{1}(\mathbb{R}^{N})}\leq\|\psi\|_{L^{1}(\mathbb{R}^{N})}$,
and hence, we obtain the stronger property

$\textup{e}^{-t}T_{t}^{-\mathfrak{L}^{\sigma,\mu}}[\psi]\to 0\qquad\text{in
$L^{1}(\mathbb{R}^{N})$ as $t\to\infty$.}$

The above can then be summarized as:

######  Proposition D.3 (The inverse operator $A^{-1}$).

  1. (a)

The semigroup $(\textup{e}^{-t}T_{t}^{-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$
associated with $(I-\mathfrak{L}^{\sigma,\mu})$, which satisfies Proposition
C.11 and Theorem C.1, has an inverse operator given, for all $0\leq\psi\in
L^{1}(\mathbb{R}^{N})$, by

$(I-\mathfrak{L}^{\sigma,\mu})^{-1}[\psi]=\int_{0}^{\infty}\textup{e}^{-t}T_{t}^{-\mathfrak{L}^{\sigma,\mu}}[\psi]\,\mathrm{d}t,$

and
$(I-\mathfrak{L}^{\sigma,\mu})^{-1}[(I-\mathfrak{L}^{\sigma,\mu})[\psi]]=\psi$.

  2. (b)

Assume that the semigroup $(T_{t}^{-\mathfrak{L}^{\sigma,\mu}})_{t\geq 0}$
associated with $(-\mathfrak{L}^{\sigma,\mu})$, which satisfies Proposition
C.11 and Theorem C.1, has a symmetric and positive kernel $\mathbb{H}_{t}$
satisfying, for all compact $K\subset\mathbb{R}^{N}$,

$\int_{0}^{\infty}\int_{K}\mathbb{H}_{t}(x)\,\mathrm{d}x\,\mathrm{d}t<\infty\qquad\text{and}\qquad\lim_{t\to\infty}\mathbb{H}_{t}(x)\to
0\quad\text{for a.e. $x\in\mathbb{R}^{N}$.}$

Then $(-\mathfrak{L}^{\sigma,\mu})$ has an inverse operator given, for all
$0\leq\psi\in L^{1}(\mathbb{R}^{N})$, by

$(-\mathfrak{L}^{\sigma,\mu})^{-1}[\psi]=\int_{0}^{\infty}T_{t}^{-\mathfrak{L}^{\sigma,\mu}}[\psi]\,\mathrm{d}t=\int_{\mathbb{R}^{N}}\bigg{(}\int_{0}^{\infty}\mathbb{H}_{t}(y)\,\mathrm{d}t\bigg{)}\psi(x-y)\,\mathrm{d}y$

and
$(-\mathfrak{L}^{\sigma,\mu})^{-1}[(-\mathfrak{L}^{\sigma,\mu})[\psi]]=\psi$.

######  Remark D.4.

  1. (a)

We immediately see that, in the setting of part (b) above,

$\mathbb{G}_{-\mathfrak{L}^{\sigma,\mu}}^{x_{0}}(x):=\int_{0}^{\infty}\mathbb{H}_{t}^{x_{0}}(x)\,\mathrm{d}t.$

Moreover, if we apply the setting of part (b) to part (a), then we also have
that

$\mathbb{G}_{I-\mathfrak{L}^{\sigma,\mu}}^{x_{0}}(x):=\int_{0}^{\infty}\textup{e}^{-t}\mathbb{H}_{t}^{x_{0}}(x)\,\mathrm{d}t.$

Of course, the latter is well-defined for a larger class of kernels than the
former.

  2. (b)

We can then check that all the examples considered in Section 7 have inverses.

The positivity assumption of (2) is then very natural since it is related to
the fact that the operator ensures the comparison principle.

######  Corollary D.5.

Under the assumptions of Proposition D.3,

$\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x),\mathbb{G}_{I-\mathfrak{L}}^{x_{0}}(x)\geq
0\qquad\text{for a.e. $x\in\mathbb{R}^{N}$.}$

###### Proof.

Since the comparison principle holds for solutions of the heat equation,
$\mathbb{H}_{-\mathfrak{L}}^{x_{0}}(x,t)\geq 0$. ∎

In fact, we only need to assume that the resolvent is nonnegative since
$\textup{e}^{-t}\leq 1$:

######  Corollary D.6.

Under the assumptions of Proposition D.3,

$0\leq\mathbb{G}_{I-\mathfrak{L}}^{x_{0}}(x)\leq\mathbb{G}_{-\mathfrak{L}}^{x_{0}}(x)\qquad\text{for
a.e. $x\in\mathbb{R}^{N}$.}$

##  Appendix E Existence and a priori results for weak dual solutions

Although this is not the main point of the paper, we will illustrate that our
assumptions (2)–(2) does not lead to an empty theory. Let us therefore prove
Proposition 4.1. To do so, we rely on the theory of abstract solutions for the
corresponding elliptic problem of (GPME). Recall Lemma 4.8, and due to (2), we
also have:

######  Lemma E.1.

Consider $A:=(-\mathfrak{L})$ or $A:=(I-\mathfrak{L})$. Then
$A[\mathbb{G}_{A}^{y}]=\delta_{y}$ in $\mathcal{D}^{\prime}(\mathbb{R}^{N})$,

$A^{-1}[\psi]=\int_{\mathbb{R}^{N}}\mathbb{G}_{A}^{0}(x-y)\psi(y)\,\mathrm{d}y,$

and $A[A^{-1}[\psi]]=\psi$ for all $\psi\in
C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$.

###### Proof of Proposition 4.1.

(a) Consider a uniform grid in time such that $0=t_{0}<t_{1}<\cdots<t_{J}=T$.
Let $\mathbb{J}:=\\{1,\ldots,J\\}$, and denote the time steps by $\Delta
t=t_{j}-t_{j-1}$ for all $j\in\mathbb{J}$. The piecewise constant time
interpolant $u_{\Delta t}$ is, for $(x,t)\in Q_{T}$, given as

$u_{\Delta t}(x,t):=u_{j}(x)\quad\text{where $t\in(t_{j-1},t_{j}]$ for all
$j\in\mathbb{J}$,}$

and $u_{\Delta t}(x,0):=u_{0}(x)$. Now, each $u_{j}$ is defined recursively as
the solution of the following elliptic equation:

$u_{j}+\Delta tA[u_{j}^{m}]=u_{j-1}\qquad\text{in $\mathbb{R}^{N}$,}$

which of course is another way of expressing (C.4). Since $A$ is densely
defined, $\mathfrak{m}$-accretive, and Dirichlet in $L^{1}(\mathbb{R}^{N})$,
the above equation has an a.e.-solution (cf. Theorem C.1 and Proposition
C.12). Then rewriting the equation, multiplying by
$A^{-1}[\psi(\cdot,t_{j-1})]$ with $\psi\in
C_{\textup{c}}^{\infty}(\mathbb{R}^{N}\times(0,T))$, integrating over
$\mathbb{R}^{N}$, using

$\int_{\mathbb{R}^{N}}A[u_{j}^{m}]A^{-1}[\psi(\cdot,t_{j-1})]=\int_{\mathbb{R}^{N}}u_{j}^{m}A[A^{-1}[\psi(\cdot,t_{j-1})]]=\int_{\mathbb{R}^{N}}u_{j}^{m}\psi(\cdot,t_{j-1}),$

and summing over $j$, we obtain that

$\sum_{j\in\mathbb{J}}\int_{\mathbb{R}^{N}}\frac{u_{j}-u_{j-1}}{\Delta
t}A^{-1}[\psi(t_{j-1})]\,\mathrm{d}x\Delta
t+\sum_{j\in\mathbb{J}}\int_{\mathbb{R}^{N}}u_{j}^{m}\psi(t_{j-1})\,\mathrm{d}x\Delta
t=0.$

We now perform summation by parts, use the symmetry of $A^{-1}$, and use that
$\psi$ has compact support in $(0,T)$ (so that it vanishes for small enough
$\Delta t$ for some $n,m\in\mathbb{J}$) to obtain

$-\sum_{j=n}^{m-1}\int_{\mathbb{R}^{N}}A^{-1}[u_{j}]\frac{\psi(t_{j})-\psi(t_{j-1})}{\Delta
t}\,\mathrm{d}x\Delta
t+\sum_{j=n}^{m-1}\int_{\mathbb{R}^{N}}u_{j}^{m}\psi(t_{j-1})\,\mathrm{d}x\Delta
t=0.$

At this point, we can follow the proof of Proposition 5.2 in [15] since, in
our case, we have that $A^{-1}[u_{j}]\in
C([0,T];L_{\textup{loc}}^{1}(\mathbb{R}^{N}))\cap L^{\infty}(Q_{T})$ and e.g.

$|u_{j}^{m}-u^{m}(t_{j})|\leq
2\|u_{0}\|_{L^{\infty}(\mathbb{R}^{N})}^{m-1}|u_{j}-u(t_{j})|.$

That is, $u\in(L^{1}\cap L^{\infty})(Q_{T})\cap
C([0,T];L^{1}(\mathbb{R}^{N}))$ satisfies

$\int_{0}^{T}\int_{\mathbb{R}^{N}}\big{(}A^{-1}[u]\partial_{t}\psi-u^{m}\psi\big{)}\,\mathrm{d}x\,\mathrm{d}t=0\qquad\text{for
all $\psi\in C_{\textup{c}}^{\infty}(Q_{T})$.}$

Assume $0<\tau_{1}\leq\tau_{2}\leq T$, and choose
$\psi(x,t)\mapsto\theta_{n}(t)\psi(x,t)$ where the new $\psi$ is in
$C_{\textup{c}}^{1}([\tau_{1},\tau_{2}];L_{\textup{c}}^{\infty}(\mathbb{R}^{N}))$
and $\theta_{n}$ is an approximation of the square pulse with support in
$[\tau_{1},\tau_{2}]$. The above expression is still well-defined for this
choice, and moreover, since e.g. $A^{-1}[u]\in
C([0,T];L_{\textup{loc}}^{1}(\mathbb{R}^{N}))$, we can take the limit as
$n\to\infty$. This concludes that $u$ is a weak dual solution according to
Definition 2.1.

(b) The comparison principle and the $L^{p}$-decay are immediately inherited
from the elliptic problem, see e.g. the proofs of Propositions 5.1 and 5.2 in
[15]. ∎

######  Remark E.2.

We have in fact shown that mild/integral solutions (i.e., the limit points of
the time-discretized problem) of (GPME) are weak dual solutions.

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