f,\phi_{t^{\prime}}\rangle|}{|R_{t^{\prime}}|^{\frac{1}{2}}}S(f)\left(\sum_{t\in\mathbb{T}}\int_{(K_{n}^{2}R_{\mathbf{T}(t)})^{\mathsf{c}}}(1+\rho_{R_{t}}(x))^{-M+n}\right)^{\frac{1}{2}},\end{split}$

where $\mathbf{T}(t)$ denotes the unique tree $\mathbf{T}\in\mathfrak{T}$ such
that $t\in\mathbf{T}$. In passing to the last line, we used the Cauchy–Schwarz
inequality together with the simple estimate

$\int_{(K_{n}^{2}R_{\mathbf{T}})^{\mathsf{c}}}(1+\rho_{R_{t}}(x))^{-M+n}\lesssim|R_{t}|,\qquad
M>2n.$

The estimate for $S_{2}$ is completed by involving the estimate

$\sum_{t\in\mathbb{T}}\int_{(K_{n}^{2}R_{\mathbf{T}(t)})^{\mathsf{c}}}(1+\rho_{R_{t}}(x))^{-M+n}=\sum_{\mathbf{T}\in\mathfrak{T}}\sum_{t\in\mathbf{T}}\int_{(K_{n}^{2}R_{\mathbf{T}})^{\mathsf{c}}}(1+\rho_{R_{t}}(x))^{-M+n}\lesssim\sum_{\mathbf{T}\in\mathfrak{T}}|R_{\mathbf{T}}|$

for $M>2n$. Balancing the estimates for $S(f)$ in terms of $S_{1}(f),S_{2}(f)$
according to which term dominates, and taking into account (5.10), yield the
desired bound. ∎

###  6.12. Selection by density

The sorting of a collection of trees by density follows exactly the standard
procedure as for example in [LTC]; we include its statement for completeness.

######  Lemma 6.13 (density lemma).

Let $\mathbb{T}\subset\mathbb{P}$ be a finite collection of tiles. There
exists a disjoint decomposition

$\mathbb{T}=\mathbb{T}^{\rm{light}}\cup\bigcup_{\tau=1}^{N}\mathbf{T}_{k}$

where $\\{\mathbf{T}_{\tau}:1\leq\tau\leq N\\}$ is a finite collection of
trees and

$\sum_{\tau=1}^{N}|R_{\mathbf{T}_{\tau}}|\lesssim\mathrm{dense}(\mathbb{T})^{-1}|E|,\qquad{\mathrm{dense}}(\mathbb{T}^{\rm{light}})\leq\frac{1}{2}\mathrm{dense}(\mathbb{T}),$

where $E$ refers to the measurable $E\subset\mathbb{R}^{d}$ in the definition
of density, with implicit constant depending only upon dimension.

###  6.14. Selection by size

We describe in the subsection the suitable version of the size (or energy)
lemma, that will eventually allows us to sort a collection of tiles into trees
of controlled size. The argument is standard in the literature but we
introduce a small variation to account for the lack of transitivity in the
relation of order that is implicit in the definition of trees. A technical
device used in order to prioritize the selection of trees appears only in our
higher dimensional setting and is defined below.

######  Definition 6.15.

(signature) Let $\mathcal{G}$ be a triadic grid of pace $(1,m)$ in
$\mathbb{R}^{d}$, $m\in[0,1)$, and $\xi\in\mathbb{R}^{d}$ having no triadic
coordinates. The _signature_ of $\xi$ is the number $\mathrm{sig}(\xi)$
defined as follows

$\mathrm{sig}(\xi)\coloneqq\sum_{j=1}^{\infty}\frac{a_{j}}{3^{j}},\qquad
a_{j}\coloneqq\begin{cases}&0,\qquad\text{if}\quad\xi\in Q^{o}\quad\text{for
some}\quad Q\in\mathcal{G}_{-j},\\\ &1,\qquad\text{if}\quad\xi\notin
Q^{o}\quad\forall Q\in\mathcal{G}_{-j},\end{cases}$

where we remember that
$\mathcal{G}_{-j}=\\{Q\in\mathcal{G},\,\ell(Q)=3^{-j+m}\\}$, see Subsection
5.2.

######  Remark 6.16.

The following point motivates the definition above: if
$\mathbf{T},\mathbf{T}^{\prime}$ are two different trees and
$t\in\mathbf{T},t^{\prime}\in\mathbf{T}^{\prime}$ with $Q_{t}\subseteq
Q_{t^{\prime}}^{\circ}$, then
$\mathrm{sig}(\xi_{\mathbf{T}})<\mathrm{sig}(\xi_{\mathbf{T}^{\prime}})$. This
fact will play a role in proving that the family of trees extracted in the
size lemma below is strongly disjoint.

######  Lemma 6.17 (size lemma).

Let $\mathbb{T}\subset\mathbb{P}$ be a finite collection of tiles. There
exists a disjoint decomposition

$\mathbb{T}=\mathbb{T}^{\rm{small}}\cup\bigcup_{\tau=1}^{N}\mathbf{T}_{\tau}$

where $\\{\mathbf{T}_{\tau}:1\leq\tau\leq N\\}$ is a finite collection of
trees and

$\sum_{\tau=1}^{N}|R_{\mathbf{T}_{\tau}}|\lesssim\mathrm{size}(\mathbb{T})^{-2}\|f\|_{L^{(}\mathbb{R}^{n})}^{2},\qquad\mathrm{size}(\mathbb{T}^{\rm{small}})\leq\frac{\mathrm{size}(\mathbb{T})}{\sqrt{2}},$

with implicit constant depending only upon dimension.

###### Proof.

Throughout the proof we abbreviate $\sigma\coloneqq\mathrm{size}(\mathbb{T})$.
We remove trees from $\mathbb{T}$ via the following recursive procedure. First
consider all the maximal (with respect to set inclusion) lacunary trees
$\mathbf{T}\subseteq\mathbb{T}$ such that

(6.2) $\sum_{t\in\mathbf{T}}F[f](t)^{2}>\frac{\sigma^{2}}{2}|R_{\mathbf{T}}|$

and among those let $\mathbf{T}_{1}$ be the one with $\xi_{\mathbf{T}_{1}}$
having the _smallest signature_ , where
$(\xi_{\mathbf{T}_{1}},R_{\mathbf{T}_{1}})$ is the top of $\mathbf{T}_{1}$. We
define the collection of tiles

$\mathcal{E}(\mathbf{T}_{1})\coloneqq\\{t\in\mathbb{T}:\,\xi_{\mathbf{T}_{1}}\in
Q_{t},\,R_{t}\cap K_{n}^{2}R_{\mathbf{T}_{1}}\neq\varnothing\\}.$

Clearly $\mathbf{T}_{1}\subseteq\mathcal{E}(\mathbf{T}_{1})$. Now replace
$\mathbb{T}$ by $\mathbb{T}\setminus\mathcal{E}(\mathbf{T}_{1})$ and recurse.
The selection algorithm terminates when $\mathrm{size}(\mathbb{T}^{\rm
small})=\mathrm{size}(\mathbb{T}\setminus\bigcup_{k}\mathcal{E}(\mathbf{T}_{k}))<C_{1}\sigma$.

Note that the collection $\\{\mathcal{E}(\mathbf{T}_{k})\\}_{k}$ are not —
strictly speaking — trees according to Definition 6.1 but we can easily remedy
that. Indeed consider the rectangular parallelepipeds $R_{\mathbf{T}_{k,j}}$
which belong to the same grid as $R_{\mathbf{T}_{k}}$ and are such that
$R_{\mathbf{T}_{k,j}}\cap K_{n}^{2}R_{\mathbf{T}_{k}}\neq\varnothing$. Clearly
there are at most $O_{d}(1)$ indices $j$ for which this happens, uniformly in
$k$. Now we set $T_{k,1}\coloneqq\mathbf{T}_{k}$ and recursively let
$\mathbf{T}_{k,j+1}$ be the maximal tree with top
$(\xi_{T},R_{\mathbf{T}_{k,j+1}})$ contained in
$\mathcal{E}(\mathbf{T}_{k})\setminus\cup_{j^{\prime}\leq
j}\mathbf{T}_{k,j^{\prime}}$. Each $\mathbf{T}_{j,k}$ is a tree and we have
$\sum_{k}\sum_{j}|R_{\mathbf{T}_{j,k}}|\lesssim\sum_{k}|R_{\mathbf{T}_{k}}|$
so the proof will be complete once we estimate the latter sum. An important
fact is that the collection $\mathfrak{T}\coloneqq\\{\mathbf{T}_{k}\\}_{k}$ is
strongly disjoint. Indeed if $t\in\mathbf{T}\neq\mathbf{T}^{\prime}\ni
t^{\prime}$ with $\mathbf{T},\mathbf{T}^{\prime}\in\mathfrak{T}$ with
$Q_{t}\subseteq Q_{t^{\prime}}^{\circ}$ and the tree $\mathbf{T}$ was selected
first because of Remark 6.16. Thus if we had that $R_{t^{\prime}}\cap
K_{n}^{2}R_{\mathbf{T}}\neq\varnothing$ then the tile $t^{\prime}$ would have
been included in the family $\mathcal{E}(\mathbf{T})$ and consequently would
not be available for inclusion in the tree $\mathbf{T}^{\prime}$. Now using
the selection condition (6.2) together with Lemma 6.11 we get

$\sum_{\mathbf{T}\in\mathfrak{T}}|R_{\mathbf{T}}|\lesssim\sigma^{-2}\sum_{\mathbf{T}\in\mathfrak{T}}\sum_{t\in\mathbf{T}}F[f](t)^{2}\lesssim\sigma^{-2}\max\left(\|f\|_{2}^{2},\sigma^{\frac{2}{3}}\left(\sum_{\mathbf{T}\in\mathfrak{T}}|R_{\mathbf{T}}|\right)^{\frac{1}{3}}\|f\|_{2}^{\frac{4}{3}}\right)$

which readily yields the desired estimate. ∎

###  6.18. The single tree estimate

The size and density lemmas combined with the discretization of the operator
in (5.28) reduce matters to the estimate for a single tree which is stated and
proved below.

######  Lemma 6.19 (tree lemma).

Let $\mathbf{T}$ be a tree. There holds

$\Lambda_{\mathbf{T}}(f,g\bm{1}_{E})\lesssim\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})|R_{T}|$

where $\mathrm{size}$ is defined with respect to $f\in
L^{\infty}_{0}(\mathbb{R}^{n})$, $\mathrm{dense}$ is defined with respect to
$E\subset\mathbb{R}^{n}$ and $g\in L^{\infty}_{0}(\mathbb{R}^{n})$ with
$\|g\|_{\infty}=1$ is arbitrary.

###### Proof.

In view of the definition of the form $\Lambda_{\mathbf{T}}$ it suffices to
prove that

(6.3) $\sum_{t\in\mathbf{T}}\left|\langle
f,\varphi_{t}\rangle\right|\left|\left\langle\vartheta_{t}\left(\cdot,\sigma(\cdot)\right),g\bm{1}_{E_{t}}\right\rangle\right|\lesssim\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})|R_{\mathbf{T}}|$

uniformly in choices of adapted families
$\\{\varphi_{t}:\,\varphi_{t}\in\mathcal{F}_{t}^{M},\,t\in\mathbf{T}\\}$ and
$\\{\psi_{t}:\,\varphi_{t}\in\mathcal{A}_{t}^{M},\,t\in\mathbf{T}\\}$, where
we remember that we have fixed $M=50n$.

We begin the proof setup with a reduction. Since the single tree estimate is
rotation invariant we can and will assume that
$\xi_{\mathbf{T}}=\Pi_{e_{n}^{\perp}}e_{n}=0\in\mathbb{R}^{d}$. Lemma 6.2 then
implies that there exists a dimensional constant $C_{n}>0$ such that for any
tile $t\in\mathbf{T}$ there exists a rectangle ${R}_{t}^{*}$ with sides
parallel to the coordinate axes, $\Pi_{e_{n}^{\perp}}({R}_{t}^{*})$ is a cube
in $\mathbb{R}^{d}$ and $R_{t}\subset R_{t}^{*}\subset C_{n}R_{t}$.
Furthermore, by e.g. splitting the tree into $O_{n}(1)$ trees we may assume
that all $R_{t}^{*}$ come from a single grid $\mathcal{L}$ in $\mathbb{R}^{n}$
that consists of rectangles with sides parallel to the axes and is such that

$\\{\Pi_{e_{n}^{\perp}}(R):R\in\mathcal{L}\\}\subseteq\mathcal{D},$

where $\mathcal{D}$ is the standard dyadic grid in $\mathbb{R}^{d}$, and that
$\Pi_{e_{n}}(R^{*})$ are dyadic intervals of length $4$. We also fix
${R}_{\mathbf{T}}^{*}\in\mathcal{L}$ with $R_{\mathbf{T}}\subseteq
R^{*}_{\mathbf{T}}\subseteq C_{n}R_{\mathbf{T}}$ and $R_{t}^{*}\subseteq
R_{\mathbf{T}}^{*}$ for all $t\in\mathbf{T}$.

We note that $\mathbf{T}^{*}\coloneqq\\{R_{t}^{*}\times
Q_{t}:\,t\in\mathbf{T}\\}$ is not — strictly speaking — a collection of tiles
according to our definition in §5.4.2. However the previously defined relation
of order is still meaningful with the same formal definition and we have that

$\mathrm{size}(\mathbf{T})\eqsim\mathrm{size}(\mathbf{T}^{*}),\qquad\mathrm{dense}(\mathbf{T})\eqsim\mathrm{dense}(\mathbf{T}^{*})$

with implicit dimensional constants, where in the definitions for
$\mathrm{size}(\mathbf{T}^{*}),\mathrm{dense}(\mathbf{T}^{*})$ above we use
$t\in\mathbf{T}^{*}$ instead of the actual tiles of $\mathbf{T}$. With these
reductions and remarks taken as understood we will henceforth write
$\mathbf{T}=\mathbf{T}^{*}$ and assume that
$\\{R_{t}:\,t\in\mathbf{T}\\}\subseteq\mathcal{L}$ with $\mathcal{L}$ being a
grid as above. Not that the uncertainty relation will change in an
inconsequential way as we will now have
$\mathrm{scl}(R_{t})\ell(Q_{t})\in[c_{n}^{-1},c_{n})$ for some dimensional
constant $c_{n}>3$.

Now let $\mathcal{P}(e_{n}^{\perp})$ be the maximal dyadic cubes $Q$ in
$\mathbb{R}^{d}$ such that $3Q\nsupseteq\Pi_{e_{n}^{\perp}}R_{t}$ for any
$t\in\mathbf{T}$; note that $\mathcal{P}(e_{n}^{\perp})$ partitions
$\mathbb{R}^{d}$. Let $\mathcal{P}(e_{n})$ be the dyadic intervals of length
$2$ on the real line and define
$\mathcal{P}\coloneqq\mathcal{P}(e_{n}^{\perp})\times\mathcal{P}(e_{n})$. We
will write $\mathrm{scl}(P)\coloneqq\ell(\Pi_{e_{n}^{\perp}}P)$. For
$P\in\mathcal{P}$ we write
$\mathbf{T}=\mathbf{T}_{P}^{+}\cup\mathbf{T}_{P}^{-}$, where

$\mathbf{T}_{P}^{-}\coloneqq\left\\{t\in\mathbf{T}:\,\mathrm{scl}(R_{t})\leq\mathrm{scl}(P)\right\\}\quad\text{and}\quad\mathbf{T}_{P}^{+}\coloneqq\left\\{t\in\mathbf{T}:\,\mathrm{scl}(R_{t})>\mathrm{scl}(P)\right\\}.$

We write the basic estimate

$\sum_{t\in\mathbf{T}}\left|\langle
f,\phi_{t}\rangle\right|\left|\left\langle\vartheta_{t}\left(\cdot,\sigma(\cdot)\right),g\bm{1}_{E_{t}}\right\rangle\right|=\sum_{P\in\mathcal{P}}\int_{P}L_{P}^{-}(x)\,{\rm
d}x+\sum_{P\in\mathcal{P}}\int_{P}L_{P}^{+}(x)\,{\rm d}x,$

where

$L_{P}^{\omega}(x)\coloneqq\sum_{t\in\mathbf{T}_{P}^{-}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\vartheta_{t}(x,\sigma(x))g(x)\bm{1}_{E_{t}}(x),\quad\omega\in\\{-,+\\},\quad
x\in\mathbb{R}^{n},$

for an appropriate sequence $\\{\varepsilon_{t}\\}_{t\in\mathbf{T}}$ on the
unit circle in $\mathbb{C}$. We shall prove that

(6.4) $\left|\sum_{P\in\mathcal{P}}\int_{P}L_{P}^{\omega}(x)\,{\rm
d}x\right|\lesssim\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})|R_{\mathbf{T}}|,\quad\omega\in\\{-,+\\}.$

#### Proof of (6.4) for $\omega=-$

For $P\in\mathcal{P}$ and $t\in\mathbf{T}_{P}^{-}$, as $\|g\|_{\infty}\leq 1$

$\begin{split}\left|\int_{P}L_{P}^{-}(x)\,{\rm
d}x\right|&\leq\mathrm{size}(\mathbf{T})\int_{P\cap
E_{t}}|R_{t}|^{1/2}\left|\vartheta_{t}\left(x,\sigma(x)\right)\right||g(x)|\,{\rm
d}x\\\
&\lesssim\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})|R_{t}|\sup_{x\in
P}\mathsf{Sy}^{\infty}_{R_{t}}\chi_{40n}(x).\end{split}$

For $R\in\\{R_{t}:\,t\in\mathbf{T}_{P}^{-}\\}$, set

$\rho_{R}(P)\coloneqq{\inf_{x\in
P}}\,\rho_{R}(x),\qquad\rho_{R}(x)\coloneqq\inf\\{r>0:\,x\in rR\\},$

and note that
$\rho_{R_{t}}(P)\geq\max(1,\rho_{R_{\mathbf{T}}}(P))\eqqcolon\rho(P,\mathbf{T})$
by the construction of $\mathcal{P}$ and the fact that $R_{t}\subseteq
R_{\mathbf{T}}$ for $t\in\mathbf{T}$. It follows that

$\begin{split}\left|\sum_{P\in\mathcal{P}}\int_{P}L_{P}^{-}(x)\,{\rm
d}x\right|&\lesssim_{N}\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})\sum_{P\in\mathcal{P}}\sum_{t\in\mathbf{T}_{P}^{-}}|R_{t}|(1+\rho_{R_{t}}(P))^{-40n}.\end{split}$

Note that for $t\in\mathbf{T}_{P}^{-}$ with
$\rho_{R_{t}}\in[2^{\ell},2^{\ell+1})$ and $\ell(R_{t})=2^{m}$ we have

$R_{t}\subseteq
2^{\ell+4}\frac{2^{m+1}}{\ell\big{(}\Pi_{e_{n}^{\perp}}(P)\big{)}}\Pi_{e_{n}^{\perp}}(P)\times
2^{\ell+4}\Pi_{e_{n}}(P).$

We can thus estimate

$\begin{split}\sum_{P\in\mathcal{P}}\sum_{t\in\mathbf{T}_{P}^{-}}|R_{t}|\rho_{t}(P)^{-40n}&\lesssim\sum_{P\in\mathcal{P}}\sum_{2^{\ell}\geq\rho(P,\mathbf{T})}2^{-40n\ell}\sum_{{2^{m}=4}}^{\ell(\Pi_{e_{n}^{\perp}}(P))}\left(\frac{2^{\ell}2^{m}}{\ell\big{(}\Pi_{e_{n}^{\perp}}(P)\big{)}}\right)^{d}2^{\ell}|P|\lesssim\sum_{P\in\mathcal{P}}\sum_{2^{\ell}\geq\rho(P,\mathbf{T})}2^{-39n\ell}|P|\\\
&=\sum_{\begin{subarray}{c}P\in\mathcal{P}\\\ P\subseteq
100R_{\mathbf{T}}\end{subarray}}\sum_{\ell\geq
0}2^{-39n\ell}|P|+\sum_{\begin{subarray}{c}P\in\mathcal{P}\\\ P\nsubseteq
100R_{\mathbf{T}}\end{subarray}}\sum_{2^{\ell}\geq\rho_{R_{\mathbf{T}}}(P)}2^{-39n\ell}|P|\eqqcolon\mathrm{I}+\mathrm{II}.\end{split}$

Since the collection $\\{P\\}_{P\in\mathcal{P}}$ is pairwise disjoint we
readily get that $\mathrm{I}\lesssim|R_{\mathbf{T}}|$. For $\mathrm{II}$ note
that $P\nsubseteq 100R_{\mathbf{T}}$ implies that
$P\subseteq(4R_{\mathbf{T}})^{\mathsf{c}}$ so that

$\mathrm{II}\lesssim\sum_{\begin{subarray}{c}P\in\mathcal{P}\\\
P\subseteq(4R_{\mathbf{T}})^{\mathsf{c}}\end{subarray}}\int_{P}\rho_{R_{\mathbf{T}}}(x)^{-39n}\,{\rm
d}x\lesssim\int_{(4R_{\mathbf{T}})^{\mathsf{c}}}\rho_{R_{\mathbf{T}}}(x)^{-39n}\,{\rm
d}x\lesssim|R_{\mathbf{T}}|$

and the proof of (6.4) for $\omega=-$ is complete.

#### Proof of (6.4) for $\omega=+$

We make the preliminary observation that if
$\mathbf{T}_{P}^{+}\neq\varnothing$ then $\mathrm{scl}(R_{\mathbf{T}})\geq
2\mathrm{scl}(P)$ and $\Pi_{e_{n}^{\perp}}P\subseteq
5\Pi_{e_{n}^{\perp}}R_{\mathbf{T}}$. For each $P\in\mathcal{P}$ we define

$G_{P}\coloneqq P\cap\bigcup_{t\in\mathbf{T}_{P}^{+}}E_{t}.$

######  Lemma 6.20.

Let $P\in\mathcal{P}$ such that $\mathbf{T}_{P}^{+}\neq\varnothing$. Then
$|G_{P}|\lesssim(1+\rho_{R_{\mathbf{T}}}(P))^{10n}\mathrm{dense}(\mathbf{T})|P|.$

###### Proof.

Let $\widehat{P}\in\mathcal{L}$ be such that
$\Pi_{e_{n}^{\perp}}(\widehat{P})$ is the dyadic parent of
$\Pi_{e_{n}^{\perp}}(P)$. By the maximality of $P$ one has
$3\Pi_{e_{n}^{\perp}}(\widehat{P})\supseteq\Pi_{e_{n}^{\perp}}(R_{t_{0}})$ for
some $t_{0}\in\mathbf{T}$. Hence, there exists an $R\in\mathcal{L}$ such that
$\ell(\Pi_{e_{n}^{\perp}}(R))=\ell(\Pi_{e_{n}^{\perp}}(\widehat{P}))$ and
$R\supseteq R_{t_{0}}$.

Let $Q$ be the largest cube in $\mathcal{G}$ such that $\xi_{\mathbf{T}}\in
Q\subseteq Q_{t_{0}}$ and
$\mathrm{scl}(R)\mathrm{scl}(Q)\in[c_{n}^{-1},c_{n})$. Define $t\coloneqq
R\times Q$ and note that $t\geq t_{0}$ and $G_{P}\subseteq P\cap E_{t}$. As

$|P|\eqsim|R|,\qquad\mathsf{Sy}^{1}_{R}\chi_{10n}(x)\gtrsim(1+\rho_{R_{\mathbf{T}}}(P))^{-10n}|R|^{-1}\qquad\forall
x\in P,$

we have

$|G_{P}|\lesssim|P|\int_{E_{t}}\frac{1}{|R|}\bm{1}_{P}(x){\rm
d}x\lesssim(1+\rho_{R_{\mathbf{T}}}(P))^{10n}|P|\int_{E_{t}}\mathsf{Sy}^{1}_{R}\chi_{10n}(x){\rm
d}x\lesssim(1+\rho_{R_{\mathbf{T}}}(P))^{10n}|P|\mathrm{dense}(\mathbf{T}),$

and the proof is complete. ∎

We can decompose
$\mathbf{T}=\mathbf{T}_{{\text{ov}}}\cup\mathbf{T}_{\text{lac}}$, where
$\mathbf{T}_{{\text{ov}}}$ is an overlapping tree and
$\mathbf{T}_{\text{lac}}$ is a lacunary tree. For $P\in\mathcal{P}$, we set
$\mathbf{T}_{P,\text{ov}}^{+}\coloneq\mathbf{T}_{\text{ov}}\cap\mathbf{T}_{P}^{+}$
and
$\mathbf{T}_{P,\text{lac}}^{+}\coloneq\mathbf{T}_{\text{lac}}\cap\mathbf{T}_{P}^{+}$.
We define

$L_{P,\rm{type}}^{+}(x)\coloneqq\sum_{t\in\mathbf{T}_{P,\rm{type}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\vartheta_{t}(x,\sigma(x))g(x)\bm{1}_{E_{t}}(x),\quad\rm{type}\in\\{\rm{ov},\rm{lac}\\},\quad
x\in\mathbb{R}^{n}.$

We shall prove the estimates

(6.5) $\displaystyle\|L_{P,\rm{ov}}^{+}\|_{L^{\infty}(P)}$
$\displaystyle\lesssim\mathrm{size}(\mathbf{T})(1+\rho_{R_{\mathbf{T}}}(P))^{-50n},$
(6.6) $\displaystyle\|L_{P,\rm{lac}}^{+}\|_{L^{\infty}(P)}$
$\displaystyle\lesssim(1+\rho_{R_{\mathbf{T}}}(P))^{-20n}\inf_{P}{\mathrm{M}}_{\mathrm{S}}\left(\sum_{t\in\mathbf{T}_{\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\widetilde{\vartheta}_{t}\right)+\mathrm{size}(\mathbf{T})(1+\rho_{R_{\mathbf{T}}}(P))^{-50n},$

where
$\\{\widetilde{\vartheta}_{t}:\,t\in\mathbf{T}_{\rm{lac}}^{+}\\}\subset\mathcal{A}_{t}^{30n}$
and ${\mathrm{M}}_{\rm{S}}$ is the strong maximal function. Before doing so
let us show that (6.5) and (6.6) imply (6.4) for $\omega=+$. Indeed, using
(6.5) and Lemma 6.20 we have

$\begin{split}\left|\sum_{P\in\mathcal{P}}\int_{P}{L^{+}_{P,\rm{ov}}}\right|&\lesssim\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})\sum_{P\in\mathcal{P}}|P|(1+\rho_{R_{\mathbf{T}}}(P))^{-40n}\lesssim\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})|R_{\mathbf{T}}|.\end{split}$

On the other hand, using (6.6) and Lemma 6.20 and the same calculation as in
the last display we get

$\begin{split}\int\left|\sum_{P\in\mathcal{P}}L_{P,\rm{lac}}^{+}(x)\,{\rm
d}x\right|&\lesssim\mathrm{size}(\mathbf{T})\mathrm{dense}(\mathbf{T})|R_{\mathbf{T}}|\\\
&\qquad+\mathrm{dense}(\mathbf{T})\sum_{P\in\mathcal{P}}(1+\rho_{R_{\mathbf{T}}}(P))^{-10n}\int_{P}{\mathrm{M}}_{\rm{S}}\left(\sum_{t\in\mathbf{T}_{\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\widetilde{\vartheta}_{t}\right).\end{split}$

Observe that

$\begin{split}\sum_{\begin{subarray}{c}P\in\mathcal{P}\\\ P\subseteq
100R_{\mathbf{T}}\end{subarray}}\int_{P}{\mathrm{M}}_{\rm{S}}\left(\sum_{t\in\mathbf{T}_{\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\widetilde{\vartheta}_{t}\right)\lesssim|R_{\mathbf{T}}|\mathrm{size}(\mathbf{T})\end{split}$

by the Cauchy–Schwarz inequality, the boundedness of the strong maximal
function on $L^{2}(\mathbb{R}^{n})$ and Lemma 6.7. On the other hand,
remembering that $\mathrm{scl}(R_{\mathbf{T}})\geq 2\mathrm{scl}(P)$ we get

$\sum_{\begin{subarray}{c}P\in\mathcal{P}\\\ P\nsubseteq
100R_{\mathbf{T}}\end{subarray}}(1+\rho_{R_{\mathbf{T}}}(P))^{-10n}\int_{P}{\mathrm{M}}_{\rm{S}}\left(\sum_{t\in\mathbf{T}_{\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\widetilde{\vartheta}_{t}\right)\lesssim\sum_{\ell\geq
2}2^{-10\ell n}\sum_{\begin{subarray}{c}P\in\mathcal{P}\\\
\rho_{R_{\mathbf{T}}}(P)\eqsim
2^{\ell}\end{subarray}}\int_{P}{\mathrm{M}}_{\rm{S}}\left(\sum_{t\in\mathbf{T}_{\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\widetilde{\vartheta}_{t}\right)\\\
\qquad\lesssim\sum_{\ell\geq 2}2^{-10\ell
n}|2^{\ell+2}R_{\mathbf{T}}|^{\frac{1}{2}}|R_{\mathbf{T}}|^{\frac{1}{2}}\mathrm{size}(\mathbf{T})\lesssim|R_{\mathbf{T}}|\mathrm{size}(\mathbf{T})$

where we also used that $\rho_{R_{\mathbf{T}}}(P)\eqsim 2^{\ell}\implies
P\subseteq 2^{\ell+2}R_{\mathbf{T}}$ in passing to the last line.

#### Proof of (6.5)

Observe that for fixed $x\in\mathbb{R}^{n}$ if, for
$t,t^{\prime}\in\mathbf{T}_{P,\text{ov}}^{+}$, $\vartheta_{t}(x,\sigma(x))\neq
0$ and $\vartheta_{t^{\prime}}(x,\sigma(x))\neq 0$ then by property (iv) one
has $\mathrm{scl}(R_{t})=\mathrm{scl}(R_{t^{\prime}})$. Hence,

(6.7)
$\|L_{P,\rm{ov}}^{+}\|_{L^{\infty}(P)}\leq\mathrm{size}(\mathbf{T})\sup_{\begin{subarray}{c}x\in
P\\\ \mathrm{scl}\in
2^{\mathbb{N}}\end{subarray}}\sum_{\begin{subarray}{c}t\in\mathbf{T}_{P,\text{ov}}^{+}\\\
\mathrm{scl}(R_{t})=\mathrm{scl}\end{subarray}}|R_{t}|^{\frac{1}{2}}|\vartheta_{t}(x,\sigma(x))||g(x)|\bm{1}_{E_{t}}(x).$

Since $\rho_{R_{t}}(P)\geq\rho_{R_{\mathbf{T}}}(P)|R_{\mathbf{T}}|/|R_{t}|$
and $\|g\|_{\infty}\leq 1$ it follows that

$\|L_{P,\rm{ov}}^{+}\|_{L^{\infty}(P)}\lesssim\mathrm{size}(\mathbf{T})(1+\rho_{R_{\mathbf{T}}}(P))^{-50n}\sup_{\mathrm{scl}\in
2^{\mathbb{N}}}\sum_{\begin{subarray}{c}t\in\mathbf{T}_{P,\text{ov}}^{+}\\\
\mathrm{scl}(R_{t})=\mathrm{scl}\end{subarray}}\frac{|R_{t}|}{|R_{\mathbf{T}}|}\leq\mathrm{size}(\mathbf{T})(1+\rho_{R_{\mathbf{T}}}(P))^{-50n}$

since the collection
$\\{R_{t}:\,t\in\mathbf{T}_{P,\text{ov}}^{+},\,\mathrm{scl}(R_{t})=\mathrm{scl}\\}$
is pairwise disjoint and $R_{t}\subseteq R_{\mathbf{T}}$ for all
$t\in\mathbf{T}$.

#### Proof of (6.6)

Remember that $\xi_{\mathbf{T}}=\Pi_{e_{n}^{\perp}}e_{n}=0\in\mathbb{R}^{d}$
which means that property (v) of §5.4.3 applies for
$\vartheta(\cdot,\sigma(\cdot))-\vartheta(\cdot,\mathbb{R}^{d})$ where
$\mathbb{R}^{d}\in\mathrm{\bf{Gr}}(d,n)$ regarded as an element of the
Grassmanian. Writing
$\vartheta_{t}\coloneqq\vartheta_{t}(\cdot,\mathbb{R}^{d})$ we preliminary
split

$\begin{split}L^{+}_{P,\rm{lac}}&=g\left(\sum_{t\in\mathbf{T}_{P,\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\vartheta_{t}\bm{1}_{E_{t}}\right)+g\left(\sum_{t\in\mathbf{T}_{P,\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle[\vartheta_{t}(\cdot,\sigma(\cdot))-\vartheta_{t}]\bm{1}_{E_{t}}\right)\\\
&\eqqcolon gL_{P,1}+gL_{P,2}.\end{split}$

As $\|g\|_{\infty}\leq 1$ it suffices to prove the required pointwise estimate
is satisfied by $L_{P,j}$, $j=1,2$. We first deal with the term $L_{P,1}$.
Using the definition of the sets $E_{t}$, cf. (6.1), we see that for fixed
$x\in P$, a term appearing in the sum defining $L_{P,1}(x)$ is nonzero if and
only $v_{\sigma(x)}\in\alpha_{t}$ so the sets $\alpha_{t}$ are nested, and so
are the sets $E_{t}$. Hence there exist positive numbers
$\mathrm{scl}(P)<m(x)\leq M(x)$ such that the condition $\bm{1}_{E_{t}}(x)\neq
0$ is satisfied for all tiles $t$ with $m(x)\leq\mathrm{scl}(R_{t})\leq M(x)$.
Letting
$t_{R_{\mathbf{T}}}\coloneqq\mathsf{Sy}_{R_{\mathbf{T}}}^{\infty}\chi_{20n}$
we then have for any $c>0$

$L_{P,1}(x)=c^{-1}t_{R_{\mathbf{T}}}(x)\bm{1}_{E}(x)\sum_{\begin{subarray}{c}t\in\mathbf{T}_{\rm{lac}}^{+}\\\
m(x)\leq\mathrm{scl}(R_{t})\leq M(x)\end{subarray}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\widetilde{\vartheta}_{t}(x)$

where $\widetilde{\vartheta}_{t}\coloneqq
ct_{R_{\mathbf{T}}}^{-1}\vartheta_{t}$. Note that for a choice of $c$
depending only upon dimension we will have that
$\\{\widetilde{\vartheta}_{t}:\,t\in\mathbf{T}^{+}_{\rm{lac}}\\}\subset\mathcal{A}_{t}^{30n}$.
In order to conclude the desired estimate for $L_{P,1}$ we consider
$\eta\in\mathcal{S}(\mathbb{R}^{n})$ with
${\mathrm{supp}\,}(\widehat{\eta})\subseteq[-3,3]^{d}\times[1/4,4]$ and
$\widehat{\eta}(\xi)=1$ for all $\xi\in[-2,2]^{d}\times[1/2,2]$. Then, as
$\mathrm{supp}\ (\widetilde{\vartheta}_{t})^{\wedge}\subseteq\omega_{t}$, we
may write for any $B>\beta>0$

$\displaystyle\sum_{\begin{subarray}{c}t\in\mathbf{T}_{\text{lac}}^{+}\\\
\beta\leq\mathrm{scl}(R_{t})\leq B\end{subarray}}\varepsilon_{t}\langle
f,\phi_{t}\rangle\widetilde{\vartheta}_{t}=\left(\eta_{\gamma_{n}B}-\eta_{\gamma_{n}\beta}\right)*\sum_{t\in\mathbf{T}_{\text{lac}}^{+}}\varepsilon_{t}\langle
f,\phi_{t}\rangle\widetilde{\vartheta}_{t}$

for a dimensional constant $\gamma_{n}>0$; we used the notation
$\eta_{r}(x)\coloneqq r^{-d}\sigma(r^{-1}\Pi_{e_{n}^{\perp}}x+\Pi_{e_{n}}x)$
for $r>0$ and $x\in\mathbb{R}^{n}$. It follows that

$\sup_{x\in
P}|L_{P,1}(x)|\lesssim(1+\rho_{R_{\mathbf{T}}}(P))^{-20n}\inf_{P}{\mathrm{M}}_{\rm{S}}\left(\sum_{t\in\mathbf{T}_{P,\rm{lac}}^{+}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\widetilde{\vartheta}_{t}\right)$

as desired.

We turn to the estimate for $L_{P,2}$. We use the fact that
$\mathrm{scl}(t)\lesssim{\rm dist}(\sigma(x),\mathbb{R}^{d})^{-1}$ for
$t\in\mathbf{T}$ together with (v) of §5.4.3 to estimate for $x\in P$

$\begin{split}|L_{P,2}(x)|&\leq\mathrm{size}(\mathbf{T})\sum_{t\in\mathbf{T}_{P,\text{lac}}^{+}}|R_{t}|^{1/2}\left|\vartheta_{t}\left(x,\sigma(x)\right)-\vartheta_{t}\left(x\right)\right|\\\
&\lesssim\mathrm{size}(\bm{\mathrm{T}}){\rm
dist}(\sigma(x),\mathbb{R}^{d})\sum_{\begin{subarray}{c}\mathrm{scl}\in
2^{\mathbb{N}}\\\ \mathrm{scl}\lesssim{\rm
dist}(\sigma(x),\mathbb{R}^{d})^{-1}\end{subarray}}\mathrm{scl}\sum_{\begin{subarray}{c}t\in\mathbf{T}_{P,\text{lac}}^{+}\\\
\mathrm{scl}(R_{t})=\mathrm{scl}\end{subarray}}\mathsf{Sy}_{R_{t}}^{\infty}\chi_{M}(x)+\\\
&\qquad+\frac{\mathrm{size}(\bm{\mathrm{T}})}{\log\left(e+{\rm
dist}(\sigma(x),\mathbb{R}^{d})^{-1}\right)}\sum_{\begin{subarray}{c}\mathrm{scl}\in
2^{\mathbb{N}}\\\ \mathrm{scl}\lesssim{\rm
dist}(\sigma(x),\mathbb{R}^{d})^{-1}\end{subarray}}\sum_{\begin{subarray}{c}t\in\mathbf{T}_{P,\text{lac}}^{+}\\\
\mathrm{scl}(R_{t})=\mathrm{scl}\end{subarray}}\mathsf{Sy}_{R_{t}}^{\infty}\chi_{M}(x)\\\
&\lesssim\mathrm{size}(\mathbf{T})(1+\rho_{R_{\mathbf{T}}}(P))^{-50n}.\end{split}$

This concludes the proof of (6.6) and with that the proof of the tree
estimate. ∎

###  6.21. The proof of (5.28)-(5.29)

Fix a finite collection $\mathbb{P}$, $g\in L^{\infty}_{0}(\mathbb{R}^{n})$
with $\|g\|_{\infty}=1$ throughout this discussion. Also fix a set
$E\subset\mathbb{R}^{n}$ of finite measure. The function $\mathrm{dense}$
below corresponds to this choice of $E$.

We first prove (5.28). For this, fix $f\in L^{\infty}_{0}(\mathbb{R}^{n})$ so
that the $\mathrm{size}$ map is computed with respect to $f$. By consecutive
applications of the size lemma, Lemma 6.17, and the density lemma, Lemma 6.13,
we can decompose the collection of tiles $\mathbb{P}$ in the form

(6.8) $\begin{split}&\mathbb{P}=\bigcup_{k\geq
K(f)}\bigcup_{\tau=1}^{N_{k}}\mathbf{T}_{k,\tau},\\\
&\mathrm{size}(\mathbf{T}_{k,\tau})\leq
2^{-\frac{k}{2}}\|f\|_{2},\qquad\mathrm{dense}(\mathbf{T}_{k,\tau})\leq\min\left\\{1,2^{-k}|E|\right\\},\qquad\sum_{\tau=1}^{N_{k}}|R_{\mathbf{T}_{k,\tau}}|\lesssim
2^{k}.\end{split}$

This is only used in the forthcoming proof of (5.29), but the value $K(f)$
above may be chosen so that

(6.9) $\mathrm{size}(\mathbb{P})\leq
2^{-\frac{K(f)}{2}}\|f\|_{2}\lesssim\|f\|_{\infty}$

by virtue of 2. from Lemma 6.7. Consequently, employing the tree Lemma 6.19

$\begin{split}\Lambda_{\mathbb{P}}(f,g\bm{1}_{E})&\lesssim\sum_{k\geq
K(f)}\sum_{\tau=1}^{N_{k}}\mathrm{size}(\mathbf{T}_{k,\tau})\mathrm{dense}(\mathbf{T}_{k,\tau})|R_{\mathbf{T}_{k,\tau}}|\lesssim\|f\|_{2}\sum_{k\in\mathbb{Z}}2^{\frac{-k}{2}}\min(1,2^{-k}|E|)\sum_{\tau=1}^{N_{k}}|R_{\mathbf{T}_{k,\tau}}|\\\
&\lesssim\|f\|_{2}|E|^{\frac{1}{2}},\end{split}$

which is the sought after estimate (5.28).

Turning to (5.29), fix $F\subset\mathbb{R}^{n}$ of finite measure.
Accordingly, in what follows, we compute the $\mathrm{size}$ with $\bm{1}_{F}$
in place of $f$. If $|F|\leq|E|$ the desired estimate (5.29) actually follows
from (5.28), as

$\Lambda_{\mathbb{P}}(\bm{1}_{F},g\bm{1}_{E})\lesssim\|\bm{1}_{F}\|_{2}|E|^{\frac{1}{2}}=|F|^{\frac{1}{p}}|F|^{\frac{1}{2}-\frac{1}{p}}|E|^{\frac{1}{2}}\leq|F|^{\frac{1}{p}}|E|^{1-\frac{1}{p}}.$

We can therefore assume that $|F|>|E|$. Then decomposition (6.8), estimate
(6.9) for $\bm{1}_{F}$ in place of $f$, and the tree lemma imply

$\quad\Lambda_{\mathbb{P}}(\bm{1}_{F},g\bm{1}_{E})\lesssim\sum_{k\in\mathbb{Z}}\min(1,2^{-\frac{k}{2}}|F|^{\frac{1}{2}})\min(1,2^{-k}|E|)2^{k}\\\
\lesssim\sum_{2^{k}\geq|F|}2^{-\frac{k}{2}}|F|^{\frac{1}{2}}|E|+\sum_{|F|>2^{k}\geq|E|}|E|+\sum_{2^{k}<|E|}2^{k}\lesssim|E|\left(1+\log\frac{|F|}{|E|}\right)\lesssim|F|^{\frac{1}{p}}|E|^{1-\frac{1}{p}}.$

whence (5.29) follows.

##  7\. Complements

###  7.1. Maximal truncations and differentiation of Besov spaces

Let $\gamma\in\mathcal{S}(\mathbb{R}^{d})$ be a radial function satisfying

(7.1) $\mathrm{supp}\widehat{\gamma}\subseteq
B^{d}(1),\qquad\widehat{\gamma}(0)=1,\qquad\|\gamma\|_{N}\coloneqq\sup_{|\alpha|,|\beta|\leq
N}\|x^{\alpha}D^{\beta}\gamma\|_{L^{\infty}(\mathbb{R}^{n})}\leq 1.$

A few minor tweaks in the proof yield that the maximally truncated version of
(1.8)

(7.2)
$T^{\star,\star}_{\mathscr{m}}f(x)\coloneqq\sup_{\sigma\in\mathcal{\mathrm{\bf{Gr}}}(d,n)}\sup_{Q\in\mathrm{SO(d)}}\sup_{h>0}\left|\int_{\mathbb{R}^{d}}m_{\sigma}(QO_{\sigma}\Pi_{\sigma}\xi)\widehat{\gamma}(h\Pi_{\sigma}\xi)\widehat{f}(\xi){\rm
e}^{2\pi i\langle x,\xi\rangle}{\rm d}\xi\right|,\quad x\in\mathbb{R}^{d},$

also satisfies the conclusions of Theorem A, under the same assumptions
therein, provided that $N$ in (7.1) is sufficiently large, depending upon
dimension. Indeed, reductions similar to those in Section 5 may be performed,
observing that the smooth cutoff at frequency scale $h^{-1}$ does not modify
adaptededness of wave packets at frequency scales $s^{-1}\lesssim h^{-1}$
while erasing the contribution of frequency scales $\gg h^{-1}$. This leads,
in analogy with (5.12), to a model sum of type

$\Lambda^{\star}_{\mathbb{P};\sigma,h,\tau,M}(f,g)=\sum_{t\in\mathbb{P}}F_{M}[f](t)A^{\star}_{\sigma,\tau,M}[g](t),$

where the coefficients $A_{\sigma,h,\tau,M}$ have been replaced by the
modified version

$A^{\star}_{\sigma,h,\tau,M}[g](t)\coloneqq\sup_{\vartheta\in\mathcal{A}_{t}^{M}}\left|\left\langle
g,\vartheta\left(\cdot,\sigma(\cdot)\right)\bm{1}_{\alpha_{t,\tau}}\left(v_{\sigma(\cdot)}\right)\bm{1}_{[\mathrm{scl}(t),\infty)}(h(\cdot))\right\rangle\right|,$

corresponding to a measurable selector function
$h:\mathbb{R}^{n}\to(0,\infty)$ and
$\sigma:\mathbb{R}^{n}\to\mathrm{\bf{Gr}}(d,n)$. All arguments of Section 6
work just as well for the modified coefficients with purely notational change:
details are left to the interested reader.

If $m_{\sigma}\equiv 1$, the maximal operator (7.2) coincides up to a constant
factor with the maximal subspace averaging operator

$T^{\star,\star}f(x)=\sup_{\sigma\in\mathcal{\mathrm{\bf{Gr}}}(d,n)}\sup_{h>0}\left|A_{{\sigma,h}}f(x)\right|,\quad
A_{\sigma,h}f(x)\coloneqq\int_{\mathbb{R}^{d}}f(x-O_{\sigma}^{-1}t)\mathrm{Dil}_{h}^{1}\gamma(t){\,{\rm
d}t},\quad x\in\mathbb{R}^{n}.$

Thus Theorem A implies the weak $(2,2)$ and strong $(p,p)$, $p>2$, bounds for
$T^{\star,\star}$, uniformly over $\gamma$ satisfying (7.1) with $N$
sufficiently large; for example $N=50(d+1)$ is sufficient. Moreover, using
scale invariance and the decay of Schwartz tails we can get that the same
boundedness property holds for $T^{\star,\star}$ if $A_{\sigma,h}$ is defined
via a Schwartz function $\gamma\in\mathcal{S}(\mathbb{R}^{d})$ satisfying

$\widehat{\gamma}(0)=1,\qquad\|\gamma\|_{100d}\leq 1;$

namely we can drop the compact Fourier support assumption. We can use these
bounds to conclude Lebesgue differentiation theorems for functions in the
nonhomogeneous Besov spaces $B_{p,1}^{s}(\mathbb{R}^{n})$, $s\geq 0$, with
norm

$\|f\|_{B_{p,1}^{s}(\mathbb{R}^{n})}\coloneqq\|Q_{0}f\|_{L^{p}(\mathbb{R}^{n})}+\sum_{k\geq
0}2^{ks}\|P_{k}f\|_{L^{p}(\mathbb{R}^{n})},$

where $Q_{0},\\{P_{k}:\,k\geq 0\\}$ is any fixed smooth nonhomogeneous
Littlewood–Paley decomposition. For the case $s>0$ we can use an alternative
characterization of Besov spaces in terms of finite differences, as for
example in [Triebel], that shows that $|f|\in B_{p,1}^{s}$ whenever $f\in
B_{p,1}^{s}$, with a corresponding inequality of the norms, and thus we can
conclude that the rough maximal averages

$M^{\star,\star}f(x)=\sup_{\sigma\in\mathcal{\mathrm{\bf{Gr}}}(d,n)}\sup_{h>0}\langle|f|\rangle_{{x,\sigma,h}},\quad\langle
f\rangle_{{x,\sigma,h}}\coloneqq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B^{n}(0,h)\cap\sigma}f(x-O_{\sigma}^{-1}t)\,{\rm
d}t,\quad x\in\mathbb{R}^{n},$

have the same boundedness properties as $T^{\star,\star}$.

######  Theorem C.

Let $\sigma:\mathbb{R}^{n}\to\mathrm{Gr}(n-1,n)$ be any measurable function
and let $2\leq p<\infty$. If $f\in B_{p,1}^{0}(\mathbb{R}^{n})$, then

$f(x)=\lim_{h\to
0^{+}}A_{\sigma(x),h}f(x)\qquad\textrm{a.e.}\,\,x\in\mathbb{R}^{n},$

whenever $A_{\sigma,h}$ is defined via a bump function with
$\widehat{\gamma}(0)=1$ and $\|\gamma\|_{100n}\leq 1$. Moreover, if $f\in
B_{p,1}^{s}$ for some $s>0$ then

$f(x)=\lim_{h\to 0^{+}}\langle
f\rangle_{{x,\sigma,h}}\qquad\textrm{a.e.}\,\,x\in\mathbb{R}^{n}.$

###### Proof.

By the comments preceeding the statement of the theorem it suffices to
consider the case $s=0$. By assumption, for each $\varepsilon>0$ fixed there
exists $k=k(\varepsilon)\geq 0$ so that

$\sum_{k>k(\varepsilon)}\|P_{k}f\|_{p}<\varepsilon.$

Let $g=Q_{0}f+\sum_{k=0}^{k(\varepsilon)}P_{k}f$. Note that $g\in
C^{\infty}(\mathbb{R}^{n})\cap L^{2}(\mathbb{R}^{n})$ and that

$\|T^{\star,\star}(f-g)\|_{p,\infty}\lesssim\sum_{k\geq
k(\varepsilon)}\|T^{\star,\star}P_{k}f\|_{p,\infty}\lesssim\sum_{k\geq
k(\varepsilon)}\|P_{k}f\|_{p}<\varepsilon$

by an application of the uniform weak-$L^{p}(\mathbb{R}^{n})$ estimate for
$T^{\star,\star}P_{k}$. The remaining part of the argument is totally
analogous to that of the proof of the Lebesgue differentiation theorem using
the Hardy–Littlewood maximal theorem and is therefore omitted. ∎

The results of Theorem C recover a corresponding result from [Mur] for Besov
spaces $B_{p,1}^{s}$ with $s>0$ and $d=1$; see also [AFN, Naibo]. The
codimension $n-d=1$ results as well as the results for zero smoothness appear
to be new.

###  7.2. Remarks on the bi-parameter problem

In this paragraph, $d$ is no longer necessarily equal to $n-1$ and we go back
to considering values $1\leq d<n$. The rotated codimension $n-d$ singular
integral (1.2), (1.6) may be naturally generalized to the bi-parameter version

$Tf(x;\sigma,Q,P)=\int_{\mathbb{R}^{n}}m(QO_{\sigma}\Pi_{\sigma}\xi,PO_{\sigma}\Pi_{\sigma^{\perp}}\xi)\widehat{f}(\xi){\rm
e}^{2\pi i\langle x,\xi\rangle}{\rm d}\xi$

associated to a bi-parameter Hörmander–Mihlin multiplier $m=m(\eta,\zeta)$ on
$\mathbb{R}^{n}=\mathbb{R}^{d}\otimes\mathbb{R}^{n-d}$, that is, satisfying
the estimates

$\sup_{0\leq|\alpha|,|\beta|\leq
A}\,\sup_{\xi\in\mathbb{R}^{d}}\sup_{\zeta\in\mathbb{R}^{n-d}}|\eta|^{|\alpha|}|\zeta|^{|\beta|}\left|D_{\eta}^{\alpha}D_{\zeta}^{\beta}m(\eta,\zeta)\right|<\infty$

and parametrized by $Q\in\mathrm{SO}(d),P\in\mathrm{SO}(n-d)$ and
$\sigma\in\mathrm{Gr}(d,n)$ or equivalently
$\sigma^{\perp}\in\mathrm{\bf{Gr}}(n-d,n)$. By arguments analogous to those in
Section 3 for the removal of the $\mathrm{SO}(d),\mathrm{SO}(n-d)$ invariances
and finite splitting, the study of the corresponding maximal operator may be
reduced to estimates for

$T^{\star}f(x)=\sup_{\sigma\in\Sigma}|Tf(x,\sigma)|,\qquad
Tf(x,\sigma)\coloneqq\int_{\mathbb{R}^{n}}m(O_{\sigma}\xi)\widehat{f}(\xi){\rm
e}^{2\pi i\langle x,\xi\rangle}{\rm d}\xi,\quad x\in\mathbb{R}^{n}$

where $\Sigma\subset\mathrm{Gr}(d,n)$ is a small
$2^{-9}\varepsilon$-neighborhood of $\mathbb{R}^{d}$. When acting on band-
limited functions, the maximal operator $T^{\star}$ may be further reduced, up
to $L^{p}$-bounded differences, to the sum of a codimension $n-d$ and a
codimension $d$ maximal singular integral operator. More specifically, let
$\\{\Gamma_{j}:j=-1,0,1\\}$ be a partition of unity on the sphere
$\mathbb{S}^{n-1}$ subordinated to the covering

$\begin{split}&\Gamma_{-1}\coloneqq\\{\xi\in\mathbb{S}^{n-1}:|\Pi_{\mathbb{R}^{d}}\xi|>1-2\varepsilon\\},\\\
&\Gamma_{0}\coloneqq\\{\xi\in\mathbb{S}^{n-1}:\varepsilon<|\Pi_{\mathbb{R}^{d}}\xi|<1-\varepsilon\\},\\\
&\Gamma_{1}\coloneqq\\{\xi\in\mathbb{S}^{n-1}:|\Pi_{\mathbb{R}^{d}}\xi|<2\varepsilon\\}.\end{split}$

and $P_{\mathsf{cn},j}$, $j=-1,0,1$ be the corresponding smooth conical
Fourier cutoff to $\Gamma_{j}$. In analogy to what was done in (5.4), the
splitting

$T^{\star}\circ P_{0}f\leq\sum_{j=-1,0,1}T^{\star}[P_{0}P_{\mathsf{cn},j}f]$

may be performed. As the singularities along $\sigma,\sigma^{\perp}$ sit away
from the support of $P_{0}P_{\mathsf{cn},j}f$ when $\sigma\in\Sigma$,
$T^{\star}[P_{0}P_{\mathsf{cn},0}f]$ may be easily controlled by the strong
maximal function. Furthermore, when $\sigma\in\Sigma$
$|\Pi_{\sigma^{\perp}}\xi|\sim|\xi|\sim 1$ on the support of
$P_{0}P_{\mathsf{cn},-1}$ while $|\Pi_{\sigma}\xi|\sim|\xi|\sim 1$ on the
support of $P_{0}P_{\mathsf{cn},1}$. A discretization procedure similar to
that of Subsection 5.7 then shows that e.g.
$T^{\star}[P_{0}P_{\mathsf{cn},-1}f]$ lies in the convex hull of the model
operators

$\mathsf{T}f(x)=\sum_{t\in\mathbb{P}}\langle
f,\varphi_{t}\rangle\psi_{t}(x,\sigma(x))\bm{1}_{\left[\frac{a}{\mathrm{scl}(t)},\frac{b}{\mathrm{scl}(t)}\right]}({\rm
dist}(\sigma(x),\sigma(t))$

for suitable constants $b>a\gg 1$, $\mathbb{P}$ is a collection of tiles $t$
whose spatial localization is a parallelepiped with $d$ sides oriented along
the subspace $\sigma(t)$ of sidelength $\mathrm{scl}(t)\gg 1$ and $n-d$ short
sides of sidelength 1 along $\sigma(t)^{\perp}$. Also, for a fixed dyadic
value of $\mathrm{scl}(t)=\mathrm{scl}$, $\sigma(t)$ varies within a
$\sim\mathrm{scl}^{-1}$ net in $\mathrm{Gr}(d,n)$ and for each fixed $\sigma$
the spatial localizations of the tiles $t\in\mathbb{P}$ with
$\sigma(t)=\sigma$ tile $\mathbb{R}^{n}$. The model for
$T^{\star}[P_{0}P_{\mathsf{cn},1}f]$ is analogous, up to exchanging
$\mathrm{Gr}(d,n)$.

When $n=2,d=1$, then $T^{\star}\circ[P_{0}P_{\mathsf{cn},\pm 1}]$ are thus
completely symmetric and both may be handled with the methods of Section 6.
When either $d$ or $n-d$ are greater than $1$, the corresponding model sums
are substantially harder. One essential reason is that their scope includes
Nikodym maximal averaging operators along subspaces of codimension $k$ larger
than one, whose critical exponent, conjectured to be $p=k+1$, is above
$L^{2}$: see [DPP_adv] for more details on this conjecture.

## References