Off-diagonal estimates for cube skeletons maximal operators

TL;DR

This paper develops off-diagonal estimates for a geometric maximal operator related to k-skeletons in Euclidean space, using interpolation and geometric analysis techniques.

Contribution

It introduces new off-diagonal bounds for skeleton-based maximal operators by combining geometric insights with interpolation methods.

Findings

Established off-diagonal estimates for skeleton maximal operators.

Connected geometric skeleton properties with harmonic analysis techniques.

Extended off-diagonal estimate methods from circular to skeleton configurations.

Abstract

We provide off-diagonal estimates for a maximal operator arising from a geometric problem of estimating the size of certain geometric configuration of k- skeletons in $\mathbb{R}^n$. This is achieved by interpolating a weak-type endpoint estimate with the known diagonal bounds. The endpoint estimate is proved by combining a geometric result about k-skeletons and adapting an argument used to prove off-diagonal estimates for the circular maximal function in the plane.

Equations

$$M_{\delta }^{k}f(x)=\underset{1\le r\le 2}{\sup }\underset{j=1}{\overset{N}{\min }}{\text{fint}}_{S_{k,\delta }^j(x,r)}|f(y)|dy,$$

$$C^{\prime }(n,k,\epsilon )\cdot {\delta }^{\frac{k-n}{2np}+\epsilon }\le {\Vert M_{\delta }^k\Vert }_{L^p\to L^p}\le C(n,k)\cdot {\delta }^{\frac{k-n}{2np}},$$

$${\Vert M_{\delta }^k\Vert }_{L^p\to L^q}=\underset{f\ne 0}{\sup }\frac{{\Vert M_{\delta }^{k}f\Vert }_{L^q}}{{\Vert f\Vert }_{L^p}}.$$

$$\begin{array}{rl}{C}^{\prime }{\delta }^{\frac{k-n}{2np}+\epsilon }\le {\Vert M_{\delta }^k\Vert }_{L^p\to L^q}& \le C{\delta }^{\frac{k-n}{2np}}\text{if}q\le q^{\ast }p\\ {\Vert M_{\delta }^k\Vert }_{L^p\to L^q}& \approx {\delta }^{\frac{n}{q}-\frac{n-k}{p}}\text{if}q>q^{\ast }p\end{array}$$

$${\Vert M_{\delta }^k\Vert }_{L^p\to L^q}\ge (N-6{)}^{n/q}{N}^{-n/p},$$

$$C_{n,k}{m}^{1-\frac{(n-k)(2n-1)}{2n^2}}$$

$${\Phi }_{\rho }(S_k(x_i,r_i))={\ell }_k^i,$$

$${\tilde{M}}_{\rho ,\delta }^{k}f(x)=\frac{1}{|{\ell }_{x,\delta }|}{\int }_{{\ell }_{x,\delta }}|f(y)|dy,$$

$${\Vert M_{\delta }^k\Vert }_{L^p\to L^q(Q_0)}\le C_{k,n}\underset{\rho \in {\Gamma }_0}{\sup }{\Vert {\tilde{M}}_{\rho ,3\delta }^k\Vert }_{L^p\to L^q(Q_0)}.$$

$$\mathrm{diam}\left(\{x\}\cup \bigcup _{r\in I}{A}_{x,r}\right)\le C$$

$$Tf(x)=\underset{r\in I}{\sup }\frac{1}{|A_{x,r}|}{\int }_{A_{x,r}}|f(y)|dy.$$

$$F=\{x\in \Omega :T{1}_E(x)>\lambda \}.$$

$$|A_{x_j,r_j}\cap E|>\lambda |A_{x_j,r_j}|,1\le j\le m.$$

$${\rm Y}=\sum _{j=1}^m{1}_{A_j^{\ast }}$$

$$|\{x\in A_j^{\ast }:{\rm Y}(x)\le \mu \}|\ge \frac{\lambda }{2}|A_j|.$$

$$\mu |E|\ge {\int }_{\{x\in E:{\rm Y}(x)\le \mu \}}{\rm Y}dx=\sum _{j=1}^m|\{x\in A_j^{\ast }:{\rm Y}(x)\le \mu \}|\ge \frac{\lambda }{2}m{\delta }^{\tau }.$$

$${\Vert Tf\Vert }_{q,\infty }\le K{\Vert f\Vert }_p,$$

$$|\{x\in \Omega :Tf(x)>t\}{|}^{1/q}\le K{t}^{-1}{\Vert f\Vert }_{L^p}.$$

$$|\{x\in \Omega :T{1}_E(x)>\lambda \}{|}^{1/q}\le C{H}^{1/p}{\delta }^{-\gamma }{\lambda }^{-1}|E{|}^{1/p},$$

$$|\{x\in \Omega :T{1}_E(x)>\lambda \}|\le c_n{\delta }^{n}m,$$

$$H^{1/p}{\delta }^{-\gamma }{\lambda }^{-1}|E{|}^{1/p}$$

$$=\frac{1}{2}{\delta }^{-\gamma +\tau /p}{m}^{\frac{1-\beta }{p}}$$

$$=\frac{1}{2}(m{\delta }^n{)}^{1/q}.$$

$$E_{{\pi }_w}:=\{x_i\in Q_0^{\ast }:{\ell }_i^k\text{ is parallel to}{\pi }_w\}.$$

$$F_{\omega }:=\{x\in {\psi }^{-1}(E_{{\pi }_{\omega }}):{\tilde{M}}_{\rho ,\delta }^k{1}_E(x)>\lambda \},$$

$$|E\cap {\ell }_{k,\delta }^j|>\lambda |{\ell }_{k,\delta }^j|.$$

$${{\rm Y}}_{\omega }=\sum _{j=1}^{m_{\omega }}{1}_{({\ell }_{\delta }^j{)}^{\ast }}$$

$$|\{x\in ({\ell }_{\delta }^j{)}^{\ast }:{{\rm Y}}_{\omega }(x)\le \mu \}|\ge \frac{\lambda }{2}|{\ell }_{\delta }^j|.$$

$${{\rm Y}}_{\omega }\le \sum _{j=1}^{m_{\omega }}{1}_{{\ell }_{\delta }^j}.$$

$${{\rm Y}}_{\omega }(x)\le C_{n,k}{m}_{\omega }^{1-\frac{(n-k)(2n-1)}{2n^2}}.$$

$$\mu \le C_{n,k}{m}_{\omega }^{1-\frac{(n-k)(2n-1)}{2n^2}}.$$

Full text

2010 Mathematics Subject Classification. Primary: 42B25. Secondary: 43A85.

The author is partially supported by grant PIP (CONICET) 11220110101018.

1. Introduction

In this work we present off-diagonal estimates for maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally over $k$-skeletons of cubes with arbitrary dimension. Roughly speaking, the $k$-dimensional boundary of an $n$-dimensional cube with axes-parallel sides in $\mathbb{R}^{n}$, for $0\leq k<n$.

The interest in this type of operators emerges, on one hand, from a geometric problem about the size of a set containing re-scaled and translated copies of the $k$-skeleton of the unit cube in $\mathbb{R}^{n}$ around every point of a set of given size. On the other hand, it is a natural variant of the celebrated spherical maximal operator of Bourgain-Stein.

The problem of finding minimal values of the size for sets in $\mathbb{R}^{n}$ containing $k$-skeletons centered at any point of a given set was introduced by Keleti, Nagy and Shmerkin [2] for the case $n=2$ and by Thornton [5], for the case $n\geq 3$. Unlike the situation of spheres, where a set containing a sphere with center in every point of a set of positive Lebesgue measure must have positive Lebesgue measure, a set containing the $(n-1)$-skeleton of an $n$-dimensional cube with axes-parallel sides and center in every point of $\mathbb{R}^{n}$ can have zero Lebesgue measure. Even more, it can have Hausdorff dimension $n-1$ (see [5]), the same as a single $(n-1)$-skeleton. This result indicates that Hausdorff dimension does not fit well with this problem. For this reason, in [2, 5], the authors obtained estimates for other notions of fractal dimensions as for example box dimension and packing dimension. The arguments from [2, 5] are direct and do not involve any maximal operators. Regardless, in [2] the authors introduced the maximal operator associated with the aforementioned geometric problem. More precisely, for each $f\in L^{1}_{\rm loc}(\mathbb{R}^{n})$, $0<\delta<1$ and $0\leq k<n\in\mathbb{N}$, the $k$-skeleton maximal operator with width $\delta$ is defined as

[TABLE]

where $S_{k,\delta}(x,r)$ is a $\delta$-neighborhood of the $k$-skeleton of a cube with center $x$ and side length $2r$, and the index $j$ enumerates its $N=2^{n-k}{n\choose k}$ faces. Here, as usual, the symbol $\fint_{E}fd\mu:=\frac{1}{\mu(E)}\int_{E}fd\mu$ denotes the average of the function $f$ with respect to the measure $\mu$ over the set $E$.

As was pointed out in [2], it turns out necessary take the minimum over all the faces of the $k$-skeleton to avoid natural and trivial results, analog to the ones for Hausdorff dimension. On the other hand, this operator cannot be bounded from $L^{p}$ to $L^{q}$ for any finite $p,q$, for otherwise a set with a $k$-skeleton centered at every point of $\mathbb{R}^{n}$ would have positive measure. Following this line, Shmerkin and the present author studied in [3] discretized versions of $M^{k}_{\delta}$, and proved nearly sharp $L^{p}$ bounds for the $k$-skeleton maximal operator. Easily one can deduce that $M^{k}_{\delta}$ is bounded on $L^{p}$, if $p>1$, just by comparison with the Hardy-Littlewood maximal operator. Nonetheless, an interesting problem is to determine the rate at which the norm of $M^{k}_{\delta}$ increases as $\delta$ goes to zero.

Theorem 1.1**.**

[3, Theorem 1.2]** Given $0\leq k<n$, $1\leq p<\infty$ and $\varepsilon>0$, there exist positive constants $C^{\prime}(n,k,\varepsilon),C(n,k)$ such that

[TABLE]

for all $\delta\in(0,1)$.

The lower bound relies on a specific construction due in [5] and to obtain the upper bound there is in fact a result for a bigger (normwise) maximal type operator which is localized on a given cube and it is linear (see Section 2 for more details). From this $L^{p}$ bound it is possible to recover the known values for the Box counting dimension of sets containing skeletons centered at any point of a prescribed set of centers under the condition of having full Box dimension (see [3, Corollary 3.5]).

Motivated by the previous result, the purpose of this article is to study the off-diagonal case. Since the $k$-skeleton maximal operator is bounded on $L^{p}$, $p\geq 1$, and is trivially bounded on $L^{\infty}$ and from $L^{1}\to L^{\infty}$, by the classical Marcinkiewicz interpolation theorem, applied to the bigger (normwise) linear maximal operator (see Definition 2.3), we can obtain the boundedness from $L^{p}\to L^{q}$, $1<p\leq q$. Nevertheless, as we mentioned before, we are interested at the rate at which the norm of the $k$-skeleton maximal operator from $L^{P}$ to $L^{q}$ increases as the parameter $\delta$ tends to 0, where as usual

[TABLE]

In the next, we say that $f(\delta)\approx g(\delta)$ if there exist two positive constants $c,c^{\prime}$, not depending on $\delta$, such that $c^{\prime}g(\delta)\leq f(\delta)\leq cg(\delta)$.

Our main result is the following:

Theorem 1.2**.**

Given $1<p\leq q<\infty$ and $\varepsilon>0$, there exist positive constants $C=C(p,q,k,n)$ and $C^{\prime}=C^{\prime}(p,q,k,n,\varepsilon)$ such that,

[TABLE]

for all $\delta\in(0,1)$ and $q^{*}=\frac{2n^{2}}{(n-k)(2n-1)}$.

The lower bound follows by applying $M^{k}_{\delta}$ to an appropriate function. For the upper bounds, we apply a combinatorial method used by Schlag in [4] for the circular maximal operator in the plane, combined with classical interpolation theorems and an estimate from the combinatorial Lemma 2.1, that we will present in the next section.

For the remaining case, $q>p$, it is straightforward conclude that the $k$-skeleton maximal operator is unbounded. In fact, given $N\in\mathbb{N}$, let $f$ be the characteristic function of an $n$-dimensional cube with side length $N$ and consider another cube, namely $N^{*}$, with the same center and side length $N-6$. By a simple calculation we obtain $M^{k}_{\delta}f(x)\geq 1$, for all $x\in N^{*}$. Therefore,

[TABLE]

which grows with $N.$

2. Preliminaries and notation

Most of the following definitions and results were introduced in [3], for completeness here we give a brief summary. In particular, we define the linear maximal operator that it turn out be the key to obtain upper bound estimates.

We denote the half-open unit cube by $Q_{0}$, i.e. $Q_{0}=[0,1)^{n}$. Let $0<\delta<1$ such that $1/\delta$ is an integer and consider the grid $Q_{0}\cap\delta\mathbb{Z}^{n}$. We define $Q_{0}^{*}:=\{x_{1},\ldots,x_{u}\}$, $u=\delta^{-n}$, the centers of the half-open $n$-cubes, $Q_{0}(1),\ldots,Q_{0}(u)$, with side length $\delta$ determined by the grid $Q_{0}\cap\delta\mathbb{Z}^{n}$.

We define the function $\psi:Q_{0}\rightarrow Q_{0}^{*}$, $x\rightarrow x^{*}$, where $x^{*}$ denotes the center of the corresponding half-open $n$-cube with center in $Q_{0}^{*}$ and side length $\delta$ containing $x$. Observe that $\psi$ is constant over each $Q_{0}(i)$ and the sets $\psi^{-1}(Q_{0}(i))$, $i=1,\ldots,u$, form a Borel partition of $Q_{0}$.

We use the letter C to denote positive constants, indicating any parameters they may depend on by subindices. Their values may change from line to line. For example, $C_{n}$ denotes a positive function of $n$.

The next combinatorial lemma is crucial in our work. It states, roughly speaking, that given a finite family of $k$-skeletons, we can extract one face from each skeleton and the overlap among them is controlled.

Lemma 2.1**.**

[3, Lemma 3.2]** There is a constant $C_{n,k}<\infty$, depending only on $n,k$, such that the following holds. Let $\{S_{k}(x_{i},r_{i})\}_{i=1}^{m}$ be a finite collection of $k$-skeletons in $\mathbb{R}^{n}$. Then it is possible to choose one $k$-face of each skeleton with the following property: If $V$ is an affine $k$-plane which is a translate of a coordinate $k$-plane, then $V$ contains at most

[TABLE]

of the chosen $k$-faces.

Definition 2.2**.**

Let $\Gamma_{0}$ denote the family of all functions $\rho:Q^{*}_{0}\rightarrow[1,2]\cap\delta\mathbb{Z}$. Fix also $\rho\in\Gamma_{0}$. For simplicity, let us write S${}_{k,i}=S_{k}(x_{i},r_{i})$, where $r_{i}=\rho(x_{i})$ and $1\leq i\leq u$.

For this family of $k$-skeletons, we define the function $\Phi_{\rho}$:

[TABLE]

where $\ell_{k}^{i}$ denotes the face of $S_{k,i}$ chosen as in Lemma 2.1.

The $k$-skeleton maximal operator defined in (1.1), unlike most other kinds maximal operators, is not sub-linear. To deal with this inconvenient we introduce a discretized and linearized version of the problem.

Definition 2.3**.**

Given a function $\rho\in\Gamma_{0}$ and $0<\delta<1$, if $f\in L^{1}_{\rm loc}(\mathbb{R}^{n})$ we define the $(\rho,k)$-skeleton maximal function with width $\delta$, ${\widetilde{M}}^{k}_{\rho,\delta}f:Q_{0}\rightarrow\mathbb{R}$ by

[TABLE]

where $\ell_{x,\delta}$ is a $\delta$-neighborhood of $\ell_{x}:=\Phi_{\rho}(S_{k}(x^{*},\rho(x^{*})))$.

Remark 2.4**.**

By definition, $\widetilde{M}^{k}_{\rho,\delta}$ is constant over each set $Q_{0}(i)$, $i=1,\ldots,u$, and it is completely determined by its values over the set $Q_{0}^{*}=\{x_{1},\ldots,x_{u}\}$.

Let $CQ_{0}$ denote the $n$-cube with the same center as $Q_{0}$ and side length $C$. Since $r$ is bounded by 2, in the previous definition it is enough to consider functions $f$ supported on $7Q_{0}$, since for each $x\in Q_{0}$, we have that $\ell_{x,\delta}\subset 7Q_{0}$.

The following result establishes the normwise relation between the $k$-skeleton maximal operator and its linearized version.

Lemma 2.5**.**

[3, Lemma 2.6]** There exists a constant $C_{k,n}>0$ such that if $0<\delta<1$,

[TABLE]

In consequence, by obtaining $L^{p}\rightarrow L^{q}$ estimates on the discrete maximal operator uniformly on $\rho$, we will also obtain $L^{p}\rightarrow L^{q}$ bounds for $M^{k}_{\delta}$, at least at a local level.

3. the weak endpoint estimate

In the present section we shall prove a weak-type $(1,q^{*})$ estimate, for the $(\rho,k)$-skeleton maximal function, the linearized version of $M^{k}_{\delta}$. To achieve this, we will follow some ideas used in [4] to treat the circular maximal operator.

Although we are interested on $k$-skeletons, first we establish the setting for more general sets.

3.1. General setting

Let $\Omega\subseteq\mathbb{R}^{n}$ be a Borel set and $I$ an index set. For each $x\in\Omega$, consider a family $\mathcal{A}_{x}=\{A_{x,r}\}_{r\in I}$ of sets with positive and finite Lebesgue measure in $\mathbb{R}^{n}$ such that $\displaystyle\inf_{r\in I}|A_{x,r}|\geq\delta^{\tau}$ for some $\tau\in\mathbb{R}$ and $0<\delta<1$. Intuitively, we can think that $\mathcal{A}_{x}$ is a family of sets at $\delta$-scale with measure equal to $\delta^{\tau}$, for all $x\in\Omega$.

In addition, suppose there exists a constant $C>0$, not depending on $x$, such that

[TABLE]

and for all $f\in L^{1}_{\rm loc}(\mathbb{R}^{n})$ consider the maximal operator $Tf:\Omega\rightarrow\mathbb{R}$ defined by

[TABLE]

Let $E\subset\mathbb{R}^{n}$ be a set with finite Lebesgue measure, $0<\lambda\leq 1$ , and $\{x_{j}\}_{j=1}^{m}$, a maximal $\delta$-separated sequence in

[TABLE]

Note that, by (3.1), $F$ is a bounded set. Pick $r_{j}\in I$ such that

[TABLE]

To simplify the notation, in the next write $A_{j}$ instead of $A_{x_{j},r_{j}}$ and $A^{*}_{j}$ instead of $A_{x_{j},r_{j}}\cap E$. Consider the multiplicity function

[TABLE]

and define $\mu$ to be the smallest integer for which there exist at least $m/2$ values of $j$ such that

[TABLE]

Observe that

[TABLE]

The following lemma characterizes the estimates on $\mu$ required to obtain restricted weak-type $(p,q)$ estimates for the maximal operator $T$.

Recall that $T$ is said to be of weak-type $(p,q)$ with norm $K$ and write

[TABLE]

if for all $f\in L^{p}(\mathbb{R}^{n})$ and $t>0\in(0,1]$,

[TABLE]

When the above inequality holds for characteristic functions of arbitrary sets in $\mathbb{R}^{n}$ of finite measure, we say that $T$ is of restricted weak-type $(p,q)$.

Lemma 3.1**.**

Let $\alpha\geq 0$ and $\beta<1$. There exists a positive constant $C=C(n,q)$ such that, if $\mu\leq H\lambda^{-\alpha}m^{\beta}$ for every choice of a bounded set $E\subset\mathbb{R}^{n}$, $0<\lambda\leq 1$ and $\delta>0$, then $T$ is of restricted weak-type $(p,q)$ with constant $CH^{1/p}\delta^{-\gamma}$, where $p=\alpha+1,q=p(1-\beta)^{-1}$ and $\gamma=\frac{\tau}{p}-\frac{n}{q}$.

Proof.

We need to prove that

[TABLE]

for every set $E\subset\mathbb{R}^{n}$ of finite measure.

Let $E\subset\mathbb{R}^{n}$ and $\{x_{j}\}_{j=1}^{m}$ be a maximal $\delta$-separated sequence in $F$, as in (3.2). Then,

[TABLE]

where $c_{n}$ denotes the measure of the $n$-dimensional ball with radius 1. In view of (3.3), i.e. $|E|\geq\mu^{-1}\frac{\lambda}{2}m\delta^{\tau}$, and by the assumption on $\mu$ we conclude that

[TABLE]

Therefore, (3.4) follows with constant $C=2c_{n}^{\frac{1}{q}}$. ∎

3.2. The case of $k$-skeletons

First, we introduce some notation and definitions.

• •

Given $n\geq 2$, $0\leq k<n\in\mathbb{N}$, and $\{e_{1},\ldots,e_{n}\}$, the canonical base in $\mathbb{R}^{n}$, we denote by $\pi_{1},\ldots,\pi_{n\choose k}$ the ${n\choose k}$ subspaces of dimension $k$ generated by vectors in the canonical base. From now on, we denote them as $k$-planes. For example, if $n=2$ and $k=1$, $\pi_{1}$ and $\pi_{2}$ are the usual coordinates axes in the plane. If $k=0$, the [math]-plane will be the origin.

• •

Let $\rho\in\Gamma_{0}$ and $\ell^{1}_{k},\ldots,\ell^{u}_{k}$ as in (2.1). We say that $\ell^{i}_{k}$, $1\leq i\leq u$, is parallel to $\pi_{\omega}$ if there exists a point $v\in\mathbb{R}^{n}$ such that it is contained in the affine subspace $V=\pi_{\omega}+v$.

• •

For each $\omega=1,\ldots,{n\choose k}$, we define the sets

[TABLE]

Let $E\subset\mathbb{R}^{n}$ be a set with finite measure, $0<\lambda\leq 1$ and consider the set

[TABLE]

where the index $\omega$ is fixed. By Remark 2.4, $F_{\omega}$ is the union of those half-open cubes $Q_{0}(i)$ with center $x_{i}\in F_{\omega}$. Note that it is enough just to consider those sets $E$ such that $E\cap 7Q_{0}\neq\emptyset$, otherwise $F_{\omega}=\emptyset$.

The set $Q^{*}_{0}\cap F_{\omega}:=\{x_{j}\}_{j=1}^{m_{\omega}}$ is a maximal $\delta$-separated set in $F_{\omega}$ and for each $x_{j}$, we have

[TABLE]

For simplicity, we write $(\ell^{j}_{\delta})^{*}$ instead of $E\cap\ell^{j}_{k,\delta}$ and $\ell^{j}_{\delta}$ instead of $\ell^{j}_{k,\delta}$.

We define the multiplicity function associated to $F_{\omega}$ by

[TABLE]

and $\mu$, as before, the smallest integer such that there exists at least $m_{w}/2$ values of $j$ such that

[TABLE]

Fix $1\leq j\leq m_{\omega}$ and consider $x\in\ell_{\delta}^{j}$. Since $(\ell^{j}_{\delta})^{*}\subseteq\ell^{j}_{\delta}$ we have that

[TABLE]

The faces $\{\ell_{k}^{1},\ldots,\ell_{k}^{m_{\omega}}\}$ were chosen from a family of $k$-skeletons using Lemma 2.1 and each one of them belongs to an affine $k$-plane parallel to $\pi_{\omega}$. Therefore, by means of the mentioned Lemma, we obtain an estimate for the number of faces containing $x$. More precisely, for all $x\in\ell_{\delta}^{j}$ we have,

[TABLE]

Since this holds for every $j=1,\ldots,m_{\omega}$, by the definition of $\mu$, we obtain

[TABLE]

The following lemma provides us the weak-type $(1,q^{*})$ estimate mentioned at the beginning of the section.

Lemma 3.2**.**

Given $1<q<\infty$, $\rho\in\Gamma_{0}$, $0<\delta<1$ and $0\leq k<n$, there exists a positive constant $C=C(k,n,q)$ such that

[TABLE]

for every $f\in L^{1}(7Q_{0})$ and $q^{*}=\frac{2n^{2}}{(n-k)(2n-1)}$.

Proof.

Consider the restricted maximal operator $\widetilde{M}^{k}_{\rho,\delta}:\psi^{-1}(E_{\pi_{\omega}})\rightarrow\mathbb{R}$. By (3.5) we have

[TABLE]

Applying Lemma 3.1 with $\alpha=0$, $\beta=1-\frac{(n-k)(2n-1)}{2n^{2}}$, $H=C_{n,k}$ and $\tau=n-k$, we obtain

[TABLE]

for every set $E\subset\mathbb{R}^{n}$ of finite measure.

Since this holds for every $\omega=1,\ldots,{n\choose k}$, and the sets $\psi^{-1}(E_{\pi_{\omega}})$ form a Borel partition of $Q_{0}$, we have

[TABLE]

and therefore, we can conclude that $\widetilde{M}^{k}_{\rho,\delta}$ is of restricted weak type $(1,q^{*})$.

If $1<q<q^{*}$,

[TABLE]

and $\widetilde{M}^{k}_{\rho,\delta}$ is of restricted weak type $(1,q)$.

For the remaining case $q>q^{*}$, take a constant $v>0$ such that $\frac{1}{q}=\frac{1}{q^{*}}-v$. Trivially by (3.5),

[TABLE]

Invoking Lemma 3.1 with $\beta=1-\frac{(n-k)(2n-1)}{2n^{2}}+v$, $\alpha=0$ and $K=C_{n,k}$ we obtain,

[TABLE]

Therefore,

[TABLE]

which implies that $\widetilde{M}^{k}_{\rho,\delta}$ is of restricted weak type $(1,q)$ with $q>q^{*}$.

Finally, by [1, Theorem 5.5.3], we can conclude that $\widetilde{M}^{k}_{\rho,\delta}$ is of weak-type $(1,q)$. ∎

4. Proof of the main theorem

In order to prove the upper bounds in Theorem 1.2, we first establish the following result.

Proposition 4.1**.**

For every $1<p\leq q$, $0<\delta<1$ and $\rho\in\Gamma_{0}$, there exist positive constants $C$ and $C^{\prime}$ depending on $k,n,p,q$ such that,

[TABLE]

Proof.

Given $(p,q)$ with $p\leq q$, by Lemma 3.2, the trivial bound

[TABLE]

and the Marcinkiewicz interpolation Theorem (See e.g.[1, Theorem 4.4.13 ]) we have the desire result. ∎

For each $z=(z_{1},\ldots,z_{n})\in\mathbb{Z}^{n}$ we denote

[TABLE]

By translation invariance, Proposition 4.1 continues to hold if we replace $Q_{0}$ by $Q_{z}$.

Proof of Theorem 1.2.

: In both cases, the upper bounds follow from Lemma 2.5, Proposition 4.1 and the facts that $\mathbb{R}^{n}=\cup_{z}Q_{z}$ and $\left\lVert\sum_{z\in\mathbb{Z}^{n}}\mathbf{1}_{7Q_{z}}\right\rVert_{\infty}$ is finite.

If $q>q^{*}p$, to obtain the lower bound, we consider $f=\mathbf{1}_{B_{6\delta}}$, where $B$ is the $k$-skeleton of an $n$-cube with side length one and center in some point $x_{0}\in\mathbb{R}^{n}$. It is easy to see that $M^{k}_{\delta}f(x)\geq 1$ for all $x$ in a $\delta$-neighborhood of $x_{0}$ and $\left\lVert M^{k}_{\delta}f\right\rVert_{L^{q}}\geq\delta^{n/q}$. Since $\left\lVert f\right\rVert_{L^{p}}=|B_{6\delta}|^{n-k/p}\approx\delta^{n-k/p}$, we have

[TABLE]

For the remaining lower bound in the case $q\leq q^{*}p$, see [3, Proposition 2.2].

∎

Acknowledgments

I thank my supervisor Pablo Shmerkin for his guidance and for his extremely valuable comments and suggestions.