#  $L^{p}-$estimates for uncentered spherical averages and lacunary maximal
functions

Ankit Bhojak  Ankit Bhojak
Department of Mathematics
Indian Institute of Science Education and Research Bhopal
Bhopal-462066, India<EMAIL_ADDRESS>,  Surjeet Singh Choudhary  Surjeet
Singh Choudhary
Department of Mathematics
Indian Institute of Science Education and Research Bhopal
Bhopal-462066, India<EMAIL_ADDRESS>,  Saurabh Shrivastava  Saurabh
Shrivastava
Department of Mathematics
Indian Institute of Science Education and Research Bhopal
Bhopal-462066, India<EMAIL_ADDRESS>and  Kalachand Shuin  Kalachand
Shuin
Research Institute of Mathematics
Seoul National University, 08826
Gwanak-RO 1, Seoul, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

We prove $L^{p}-$estimates for lacunary maximal functions associated with
uncentered spherical averages and Nikodym averages associated with spheres. In
the process, we require to prove $L^{p}-$improving properties for single scale
averaging operators in this context. In the second half of the paper, we
define bilinear analogues of uncentered spherical maximal function and the
Nikodym maximal function and study their $L^{p}-$boundedness properties.
Interestingly, in the bilinear setting, one-dimensional case is also
meaningful. We obtain $L^{p}-$estimates for lacunary uncentered bilinear
maximal functions in all dimensions.

###### 2010 Mathematics Subject Classification:

Primary 42B25; Secondary 42B15;

###### Contents

  1. 1 Introduction
  2. 2 Uncentered spherical averages and maximal functions
  3. 3 Nikodym maximal functions associated with sphere
  4. 4 Bilinear maximal functions
  5. 5 Proof of Theorem 2.4: Lacunary uncentered spherical maximal operator $N^{T}_{lac}$
  6. 6 Proof of Theorem 2.7: $L^{p}-$improving properties of $A^{T}$
  7. 7 Proof of Theorems 3.1 and 3.2: Nikodym maximal operators
  8. 8 Proof of Theorem 4.1: Lacunary bilinear maximal operator $\mathcal{N}_{lac}^{T}$ in dimension $d\geq 2$
  9. 9 Proof of Theorem 4.2: Lacunary bilinear maximal operator $\mathcal{N}_{lac}^{T}$ in dimension $d=1$
  10. 10 Necessary conditions

##  1\. Introduction

This article is motivated by recent advances in the field of spherical
averages and the corresponding maximal functions in linear and bilinear
settings. The recent works by Chang, Dosidis and Kim
[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres] on uncentered spherical
averages and by Jeong and Lee
[MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators]
for bilinear spherical maximal averages play key roles in this paper. In the
former, the authors introduced two generalizations of spherical averages and
corresponding maximal functions by considering uncentered spherical averages
and Nikodym type maximal averages. They proved $L^{p}-$estimates for these
operators. In this paper, we complement these results by proving off-diagonal
$L^{p}-$estimates for single scale averages and $L^{p}-$estimates for the
corresponding lacunary maximal averages. Further, we extend the notion of
uncentered spherical averages and Nikodym type maximal averages to bilinear
setting in the spirit of bilinear spherical maximal functions studied in Jeong
and Lee
[MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators].
Since the introductory material on each topic requires new notation, we
dedicate separate sections to describe the results on each topic mentioned
above along with the necessary preliminary background material.

### Organization of the paper

The rest of the paper is organized as follows.

  * •

In Section 2 we discuss some known results for uncentered spherical maximal
functions. Later, we describe our results on $L^{p}-$improving properties and
lacunary maximal functions in this direction. These results are proved in
Sections 5–6.

  * •

Section 3 is devoted to discuss the Nikodym maximal function associated with
spheres. In this section we state our result on the lacunary Nikodym maximal
function and bilinear Nikodym maximal function and prove them in Section 7.

  * •

The bilinear analogues of the results discussed in Sections 2 and 3 are
described in Section 4. Their proofs are discussed in Sections 8–9.

  * •

Finally, in Section 10 we discuss some examples to obtain necessary conditions
on various parameters.

##  2\. Uncentered spherical averages and maximal functions

Let $f\in\mathcal{S}(\mathbb{R}^{d}),d\geq 2$. Given $u\in\mathbb{R}^{d},$ the
uncentered spherical average of $f$ is defined by

$A^{u}_{t}f(x)=\int_{\mathbb{S}^{d-1}}f(x+t(u+y))~{}d\sigma(y),~{}t>0,$

where $d\sigma$ is the normalized surface measure on the sphere
$\mathbb{S}^{d-1}$. For $t=1$, we use the notation $A^{u}_{1}=A^{u}.$ Note
that if $u=0,A^{0}_{t}f$ is the standard spherical average. The spherical
averages are well-studies in the literature in various contexts. For example,
it is well-known that the spherical average $A^{0}_{t}f$ appears as a solution
to the wave equation. Littman [Littman] investigated $L^{p}\rightarrow
L^{q}-$estimates of the operator $A^{0}$ for $d\geq 2$. He proved the
following $L^{p}-$improving estimates for spherical averages:

(2.1)
$\displaystyle\|A^{0}f\|_{L^{q}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{d})}$

for $(1/p,1/q)$ belonging to the triangle with vertices
$\\{(0,0),(1,1),(\frac{d}{d+1},\frac{1}{d+1})\\}$. Here, the notation
$A\lesssim B$ means that there exists a constant $C>0$ such that $A\leq CB$.
We also refer the reader to [ImprovingInequalitiesForDiscreteSpherical] and
[SparseBoundsForSphericalMaximalFunction] for $L^{p}-$improving properties of
spherical averages. The corresponding $L^{p}-$improving estimates for
$A^{0}_{t}$ follows by scaling.

Recently, Chang, Dosidis and Kim
[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres] introduced maximal
functions associated with uncentered spherical averages and proved interesting
results. Given a compact set $T\subset\mathbb{R}^{d},$ they considered the
maximal function

$A^{T}f(x):=\sup_{u\in T}|A^{u}f(x)|$

and the full maximal function associated with uncentered spherical averages,
defined by

$S^{T}f(x):=\sup_{u\in T}\sup_{t>0}|A^{u}_{t}f(x)|.$

Observe that if $T=\\{0\\}$, the maximal operator $S^{T}$ corresponds to the
classical spherical maximal function, denoted by $M_{full}$. The spherical
maximal operator $M_{full}$ has been studied extensively. Stein
[SteinMaximalFunctionsSpherical] and Bourgain [BourgainCircular] proved sharp
$L^{p}-$estimates for the spherical maximal function for $d\geq 3$ and $d=2$
respectively. The $L^{p}-$estimates for $M_{full}$ holds in the range
$p>\frac{d}{d-1},$ which is known to be optimal. If we restrict the supremum
in the definition of $M_{full}$ to lacunary sequences, the resulting maximal
function is commonly referred to as the lacunary spherical maximal function
and is denoted by $M_{lac}$. The range of $p$ for which $M_{lac}$ satisfies
$L^{p}-$estimates extends to $p>1.$ We refer the reader to
[SparseBoundsForSphericalMaximalFunction] for recent developments on operators
$M_{lac}$ and $M_{full}$.

Chang, Dosidis and Kim [NikodymSetsAndMaximalFunctionsAssociatedWithSpheres]
showed that in order to get non-trivial $L^{p}-$estimates for operators
$A^{T}$ and $S^{T}$, we require suitable assumptions on ‘size’ of the
underlying set $T$. For example, if we consider $T=\mathbb{S}^{d-1},$ then the
operator $A^{\mathbb{S}^{d-1}}$ is not bounded in $L^{p}({\mathbb{R}}^{n})$
for any $p<\infty$. This assertion follows from the existence of Nikodym sets
for spheres, see[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres] and
references therein for precise details. Let us recall the notion of Nikodym
sets in the context of lines and spheres.

######  Definition 2.1.

[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres,
MattilaGeometryOfSetsMeasureEucli]

  1. (1)

Nikodym sets for lines: A Nikodym set for lines is a set
$A\subset\mathbb{R}^{d}$ of Lebesgue measure zero such that for every
$x\in\mathbb{R}^{d},$ there is a line $\tau$ which passes through $x$ and
$A\cap\tau$ contains a unit line segment.

  2. (2)

Similarly, a Nikodym set for spheres (resp., unit spheres) is a set
$T\subset{\mathbb{R}}^{d}$ of Lebesgue measure zero such that for every $y$ in
a set of positive Lebesgue measure, there exists a sphere (resp., unit sphere)
$S$ containing $y$ such that $A\cap S$ has positive $(d-1)-$dimensional
Hausdorff measure.

Nikodym sets are closely related with Kakeya sets and Besicovitch sets. The
interested reader is referred to Mattila [MattilaGeometryOfSetsMeasureEucli]
for more details about these sets. Note that the Nikodym sets for spheres in
${\mathbb{R}}^{d}$ are small with respect to the $d-$ dimensional Lebesgue
measure. However, their Hausdorff dimension must be $d$, see
[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres] for details.

In [NikodymSetsAndMaximalFunctionsAssociatedWithSpheres], Chang, Dosidis and
Kim studied $L^{p}-$boundedness of maximal functions $A^{T}$ and $S^{T}$ under
suitable conditions on the size of $T$ in terms of Minkowski content. A
compact set $T\subset\mathbb{R}^{d}$ is said to have finite $s$-dimensional
upper Minkowski content if for all $\delta\in(0,\frac{1}{2})$,

$N(T,\delta)\lesssim\delta^{-s},$

where $N(T,\delta)$ denotes the minimal number of balls of radius $\delta$
needed to cover $T.$ They proved the following result.

######  Theorem 2.2
([NikodymSetsAndMaximalFunctionsAssociatedWithSpheres],Theorem 1.7).

Let $d\geq 2$ and $0\leq s<d-1$. Let $T\subset\mathbb{R}^{d}$ be a compact set
with finite $s$-dimensional upper Minkowski content. Then $A^{T}$ is bounded
from $L^{p}(\mathbb{R}^{d})$ to itself in each of the following cases.

  1. (i)

when $d=2$ and $p>1+s$,

  2. (ii)

when $d=3$ and

$p>1+\min\Big{(}\frac{s}{2},\frac{1}{3-s},\frac{5-2s}{9-4s}\Big{)},$

  3. (iii)

when $d\geq 4$ and

$p>1+\min\Big{(}\frac{s}{d-1},\frac{1}{d-s},\frac{d-s}{3(d-s)-2}\Big{)}.$

The following result is known for the operator $S^{T}$.

######  Theorem 2.3 ([NikodymSetsAndMaximalFunctionsAssociatedWithSpheres],
Theorem $1.10$).

Let $T$ be a compact subset of $\mathbb{R}^{d}$ with upper Minkowski content
$0<s<d-1$,

  1. (i)

when $d=2$ and $0<s<1$, $S^{T}$ maps $L^{p}(\mathbb{R}^{d})$ to
$L^{p}(\mathbb{R}^{d})$ for $p>2+\min\\{1,\max(s,\frac{4s-2}{2-s})\\}$,

  2. (ii)

when $d\geq 3$ and $0<s<d-1$, $S^{T}$ maps $L^{p}(\mathbb{R}^{d})$ to
$L^{p}(\mathbb{R}^{d})$ for $p>1+[d-1-s+\max(0,\min(1,(2s-d+3)/4))]^{-1}$.

In the same paper, the authors have obtained certain necessary conditions on
$p$ for the $L^{p}-$boundedness of $S^{T}$ to hold. These necessary conditions
are sharp for a few cases.

###  2.1. Results on lacunary maximal function $N^{T}_{lac}$

Motivated by the discussion above, we consider the lacunary analogue of
maximal averages and show that the range of $p$ in Theorem 2.3 could be
improved significantly for the lacunary uncentered spherical maximal function.
This is defined as follows

$N^{T}_{lac}f(x):=\sup_{k\in\mathbb{Z}}\Big{|}A^{T}_{2^{k}}f(x)\Big{|}=\sup_{u\in
T,k\in\mathbb{Z}}\Big{|}\int_{\mathbb{S}^{d-1}}f(x+2^{k}(u+y))d\sigma(y)\Big{|}.$

Once again, observe that if $T=\\{0\\},$ the operator $N^{0}_{lac}$ coincides
with the classical lacunary spherical maximal function $M_{lac}.$ We would
like to refer the reader to Calderón [CalderonLacunarySphericalMeans],
Coifman-Weiss [CoifmanWeissBookReview], Lacey
[SparseBoundsForSphericalMaximalFunction] for $L^{p}-$estimates of
$N^{0}_{lac}=M_{lac}$ for $1<p\leq\infty$. Also, see Seeger, Tao and Wright
[EndpointMappingPropertiesOfSphericalMaximalOperators] for weak-type
$L\log\log L-$estimate for the operator $M_{lac}.$ We have the following
result for the lacunary maximal function $N^{T}_{lac}$.

######  Theorem 2.4.

Let $d\geq 2$ and $0\leq s<d-1$. Suppose that $T\subset{\mathbb{R}}^{d}$ is a
compact set with finite $s-$dimensional upper Minkowski content. Then
$N^{T}_{lac}$ is bounded on $L^{p}({\mathbb{R}}^{d})$ for each of the
following cases.

  1. (i)

$d=2$ and $p>1+s,$

  2. (ii)

$d=3$ and $p>1+\min\left(\frac{s}{2},\frac{1}{3-s},\frac{5-2s}{9-4s}\right),$

  3. (iii)

$d\geq 4$ and
$p>1+\min\left(\frac{s}{d-1},\frac{1}{d-s},\frac{d-s}{3(d-s)-2}\right).$

  4. (iv)

Moreover, $N_{lac}^{T}$ is of restricted weak-type $(p,p)$ at the endpoint
$p=1+\min\left(\frac{s}{d-1},\frac{1}{d-s}\right)$.

######  Remark 2.5.

Observe that the range of $p$ in Theorems 2.2 and 2.4 are the same.

######  Remark 2.6.

Theorem 2.4 implies the pointwise a.e. convergence of the averaging operator
$A^{T}_{2^{k}}f$, see Section $7$ in
[OnPointwisea.e.ConvergenceOfMultilinearOperators] for the necessary details.

###  2.2. Results on $L^{p}-$improving properties of $A^{T}$

We extend the $L^{p}-$estimates of $A^{T}$ to the off-diagonal range. These
are referred to as $L^{p}-$improving estimates for the operator $A^{T}$. This
completes the picture of $L^{p}\to L^{q}-$estimates for the operator $A^{T}$
with $p\leq q.$ We would like to point out that $L^{p}-$improving estimates
for averaging operators play a crucial role in proving sparse domination of
the corresponding maximal operators. For example, see
[SparseBoundsForSphericalMaximalFunction] for this connection in the context
of classical spherical maximal functions. We also refer to [RSS,
Palssonbilinearsparse, BFOPZ] for sparse domination of bilinear spherical
maximal functions. However, we do not pursue this direction of sparse
domination of maximal operators in this paper.

We need to introduce some notation in order to state the $L^{p}-$improving
results.

Let $\Delta$ denote the closed triangle with vertices $A,H$ and $E$, where
$A=(0,0)$, $E=\left(\frac{d-s}{d-s+1},\frac{1}{d-s+1}\right)$ and

  * •

$H=\Big{(}\frac{1}{1+s},\frac{1}{1+s}\Big{)},$ if $d=2$,

  * •

$H=\left(\left(1+\min\\{\frac{s}{2},\frac{1}{3-s},\frac{5-2s}{9-4s}\\}\right)^{-1},\left(1+\min\\{\frac{s}{2},\frac{1}{3-s},\frac{5-2s}{9-4s}\\}\right)^{-1}\right),$
if $d=3$,

  * •

$H=\left(\left(1+\min\\{\frac{s}{d-1},\frac{1}{d-s},\frac{d-s}{3(d-s)-2}\\}\right)^{-1},\left(1+\min\\{\frac{s}{d-1},\frac{1}{d-s},\frac{d-s}{3(d-s)-2}\\}\right)^{-1}\right),$
if $d\geq 4$.

Observe that, if $d\geq 4$ and $d-2<s<d-1,$ then
$\min\\{\frac{s}{d-1},\frac{1}{d-s},\frac{d-s}{3(d-s)-2}\\}=\frac{d-s}{3(d-s)-2}.$
Similarly, if $d=3$ and $3/2<s<2$, then
$\min\\{\frac{s}{2},\frac{1}{3-s},\frac{5-2s}{9-4s}\\}=\frac{5-2s}{9-4s}.$
With the notation as above, we have the following $L^{p}-$improving properties
of $A^{T}.$

######  Theorem 2.7.

Let $T\subset{\mathbb{R}}^{d}$ be a compact set with finite $s$-dimensional
upper Minkowski content for $0\leq s<d-1$. Then $A^{T}$ is bounded from
$L^{p}(\mathbb{R}^{d})$ into $L^{q}(\mathbb{R}^{d})$ for each of the following
cases.

  1. (i)

when $d=2$ and $(\frac{1}{p},\frac{1}{q})\in\Delta\setminus\\{E,H\\}$,

  2. (ii)

when $d=3$, $s\leq 3/2$ and
$(\frac{1}{p},\frac{1}{q})\in\Delta\setminus\\{E,H\\}$, and for $3/2<s<2$ and
$(\frac{1}{p},\frac{1}{q})\in\Delta\setminus[E,H]$,

  3. (iii)

when $d\geq 4$, $s\leq d-2$ and
$(\frac{1}{p},\frac{1}{q})\in\Delta\setminus\\{E,H\\}$, and for $d-2<s<d-1$
and $(\frac{1}{p},\frac{1}{q})\in\Delta\setminus[E,H]$.

Moreover, $A^{T}$ is of restricted weak-type $({p},{q})$, i.e., it is bounded
from the Lorentz space $L^{p,1}({\mathbb{R}}^{d})$ into
$L^{q,\infty}({\mathbb{R}}^{d}),$ for each of the following end-point values
of $p$ and $q$.

  1. (i)

when $d=2$ and $(\frac{1}{p},\frac{1}{q})=E,H$,

  2. (ii)

when $d=3$, $s\leq 3/2$ and $(\frac{1}{p},\frac{1}{q})=E,H$, and $3/2<s<2$ and
$(\frac{1}{p},\frac{1}{q})=E$,

  3. (iii)

when $d\geq 4$, $s\leq d-2$ and $(\frac{1}{p},\frac{1}{q})=E,H$, and
$d-2<s<d-1$ and $(\frac{1}{p},\frac{1}{q})=E$.

The following remarks point out the sharpness of some indices in Theorem 2.7.

######  Remark 2.8.

For $0<s\leq 1$, the restricted weak-type estimate at the point $H$ is sharp
in the sense that $A^{T}$ does not map $L^{p,r}({\mathbb{R}}^{d})$ into
$L^{q,\infty}({\mathbb{R}}^{d})$ boundedly for any $r>1$ and
$(\frac{1}{p},\frac{1}{q})=H$. We will provide an example in Section 10 to
support this assertion.

######  Remark 2.9.

The point $E$ in Theorem 2.7 is sharp. This can be verified by taking
$T=\\{0\\}^{d-\lceil s\rceil}\times C_{s},$ where
$C_{s}\subset{\mathbb{R}}^{\lceil s\rceil}$ is a self similar $s$-dimensional
set. By taking $f=\chi_{B(0,\delta)}$ for a small $\delta>0,$ we can show that
the operator $A^{T}$ is unbounded from $L^{p}({\mathbb{R}}^{d})$ to
$L^{q}({\mathbb{R}}^{d})$ if

$\frac{d}{(d-1)p}-\frac{1-s}{(d-1)q}>1.$

This shows the sharpness of the point $E$.

##  3\. Nikodym maximal functions associated with sphere

The study of Kakeya and Nikodym maximal functions is a classical topic in
harmonic analysis and geometric measure theory. These objects have been
greatly studied to understand some of the most interesting phenomena in
harmonic analysis, which includes the Fourier restriction conjecture and the
Bochner-Riesz problem. We refer the reader to
[TheKakeyaMaximalFunctionAndTheSphericalSummationMultipliers,
BesicovitchTypeMaximalOperatorsAndApplicationsToFourierAnalysis,
MattilaGeometryOfSetsMeasureEucli] for more details on these maximal
functions. Recently, in [NikodymSetsAndMaximalFunctionsAssociatedWithSpheres],
the authors introduced the Nikodym maximal function associated with spheres
and studied its $L^{p}-$boundedness properties. In this article, we are
concerned with the lacunary analogues of these maximal functions in linear and
bilinear setting.

###  3.1. Linear Nikodym maximal function $N^{\delta}$:

Let $0<\delta<1/2$, the Nikodym maximal function $N^{\delta}$ associated with
sphere $\mathbb{S}^{d-1}$ is defined by

$N^{\delta}f(x):=\sup_{u\in\mathbb{S}^{d-1}}\frac{1}{|{S}^{\delta}(0)|}\Big{|}\int_{{S}^{\delta}(0)}f(x+u+y)~{}dy\Big{|},$

where ${S}^{\delta}(0)$ denotes the $\delta$\- neighborhood of the unit sphere
$\mathbb{S}^{d-1}$. The $L^{p}-$estimates for the operator $N^{\delta}$ are
studied in [NikodymSetsAndMaximalFunctionsAssociatedWithSpheres].

Consider the corresponding lacunary maximal function defined by

$N^{\delta}_{lac}f(x):=\sup_{u\in\mathbb{S}^{d-1}}\sup_{k\in\mathbb{Z}}\frac{1}{|{S}^{\delta}(0)|}\Big{|}\int_{{S}^{\delta}(0)}f(x+2^{k}(u+y))~{}dy\Big{|}.$

We have the following bounds for the operator $N^{\delta}_{lac}$.

######  Theorem 3.1.

Let $0<\delta<1/2$ and $\epsilon>0$. Then the operator $N^{\delta}_{lac}$
satisfies the following estimates

  1. (i)

For $d=2$ we have

$\|N^{\delta}_{lac}\|_{p}\lesssim\begin{cases}\delta^{1-\frac{2}{p}-\epsilon}\|f\|_{p}&\text{if
}1<p\leq 2,\\\ \delta^{-\epsilon}\|f\|_{p}&\text{if }2\leq
p<\infty.\end{cases}$

  2. (ii)

For $d=3$ we have

$\|N^{\delta}_{lac}\|_{p}\lesssim\begin{cases}\delta^{\frac{3}{2}-\frac{5}{2p}-\epsilon}\|f\|_{p}&\text{if
}1<p\leq\frac{3}{2},\\\
\delta^{\frac{1}{2}-\frac{1}{p}-\epsilon}\|f\|_{p}&\text{if }\frac{3}{2}<p\leq
2,\\\ \delta^{-\epsilon}\|f\|_{p}&\text{if }2\leq p<\infty.\end{cases}$

  3. (iii)

Finally, for $d\geq 4$ we have

$\|N^{\delta}_{lac}\|_{p}\lesssim\begin{cases}\delta^{1-\frac{2}{p}-\epsilon}\|f\|_{p}&\text{if
}1<p\leq\frac{4}{3},\\\ \delta^{1-\frac{2}{p}-\epsilon}\|f\|_{p}&\text{if
}\frac{4}{3}<p\leq 2,\\\ \delta^{-\epsilon}\|f\|_{p}&\text{if }2\leq
p<\infty.\end{cases}$

###  3.2. Bilinear Nikodym maximal function $\mathcal{N}^{\delta}$:

Finally, we consider the bilinear analogue of the Nikodym maximal function
$N^{\delta}$ defined by

$\mathcal{N}^{\delta}(f,g)(x):=\sup_{u,v\in\mathbb{S}^{d-1}}\frac{1}{|S^{\delta}(0)|}\left|\int_{S^{\delta}(0)}f(x+u+y)g(x+v+z)\;d(y,z)\right|,$

where $0<\delta<\frac{1}{2}$ and as earlier
$S^{\delta}(0)=\\{(y,z)\in{\mathbb{R}}^{2d}:\;1-\delta<|(y,z)|<1+\delta\\}.$

We prove the following $L^{p}-$estimates for the operator
$\mathcal{N}^{\delta}.$

######  Theorem 3.2.

Let $1\leq p_{1},p_{2},p\leq\infty$ be such that
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}.$ Then

  1. (i)

$\|\mathcal{N}^{\delta}\|_{L^{p_{1}}\times L^{p_{2}}\to L^{p}}\lesssim 1$ if
$(p_{1},p_{2})\neq(1,1).$

  2. (ii)

$\|\mathcal{N}^{\delta}\|_{L^{1}\times L^{1}\to
L^{\frac{1}{2}}}\lesssim\delta^{-1}.$

##  4\. Bilinear maximal functions

###  4.1. Bilinear uncentered spherical maximal functions

Let $T\subset{\mathbb{R}}^{d},d\geq 1,$ be a compact set with finite
$s$-dimensional upper Minkowski content, where $0\leq s\leq d$. Observe that
here we allow $d=1$ as well. Define the uncentered bilinear spherical average
of dyadic scale $2^{k}$ associated with $T$ by

$\mathcal{N}_{2^{k}}^{T}(f,g)(x):=\sup_{u,v\in
T}\left|\int_{\mathbb{S}^{2d-1}}f(x+2^{k}(u+y))g(x+2^{k}(v+z))\;d\sigma_{2d-1}(y,z)\right|.$

Note that this can be rewritten in terms of Fourier transform as follows

$\mathcal{N}_{2^{k}}^{T}(f,g)(x)=\sup_{u,v\in
T}\Big{|}\int_{\mathbb{R}^{2d}}\widehat{\sigma}_{2d-1}(2^{k}\xi,2^{k}\eta)e^{2\pi\iota
2^{k}(u\cdot\xi+v\cdot\eta)}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi\iota
x\cdot(\xi+\eta)}~{}d\xi d\eta\Big{|},$

where $\widehat{\sigma}_{2d-1}$ denotes the Fourier transform of surface
measure of the sphere $\mathbb{S}^{2d-1}$. It is well-known that the pointwise
a.e. convergence

$\lim_{k\rightarrow-\infty}\mathcal{N}_{2^{k}}^{T}(f,g)(x)=\widehat{\sigma}_{2d-1}(0)f(x)g(x)~{}~{}\text{and}~{}~{}\lim_{k\rightarrow+\infty}\mathcal{N}_{2^{k}}^{T}(f,g)(x)=0$

can be deduced by proving appropriate $L^{p}-$estimates for the corresponding
lacunary maximal function defined by

$\mathcal{N}_{lac}^{T}(f,g)(x):=\sup_{k\in\mathbb{Z}}\;\mathcal{N}_{2^{k}}^{T}(f,g)(x).$

Observe that if $T=\\{\vec{0}\\},$ the maximal function
$\mathcal{N}_{lac}^{0}$ reduces to the classical bilinear lacunary spherical
maximal function

$\mathcal{S}_{lac}(f,g)(x):=\sup_{k\in\mathbb{Z}}\left|\int_{\mathbb{S}^{2d-1}}f(x+2^{k}y)g(x+2^{k}z)\;d\sigma_{2d-1}(y,z)\right|.$

The operator $\mathcal{S}_{lac}$ is well studied. For $d=1,$ Christ and Zhou
[ChristZhou] proved $L^{p_{1}}({\mathbb{R}})\times
L^{p_{2}}({\mathbb{R}})\rightarrow L^{p}({\mathbb{R}})-$boundedness of this
operator for almost complete range of exponents $p_{1},p_{2}$ and $p$. We also
for refer to [DosidisRamos] for $L^{p}-$ boundedness of the continous analogue
of $\mathcal{S}_{lac}$ and [BCSS] for its sharp endpoints bounds in dimension
one. Recently, Borges and Foster [Borges] and Cho, Lee and Shuin [CLS] have
proved $L^{p_{1}}({\mathbb{R}}^{d})\times
L^{p_{2}}({\mathbb{R}}^{d})\rightarrow L^{p}({\mathbb{R}}^{d})-$boundedness of
$\mathcal{S}_{lac}$ for $d\geq 2,$ except for border line cases. Also see
[LeeShuin] for bilinear maximal functions defined on degenerate surfaces.

We extend these results to the context of the lacunary maximal function
$\mathcal{N}_{lac}^{T}$. Given a set of point $\\{X_{1},X_{2},\dots,X_{k}\\}$
in ${\mathbb{R}}^{2}$, let $\Omega(\\{X_{1},X_{2},\dots,X_{k}\\})$ denote the
open convex hull of all the points $X_{1},X_{2},\dots,X_{k}.$ We define the
points $O=(0,0)$, $A=(1,0)$, and $B=(0,1)$. The points $P=P(d,s),\;Q=Q(d,s)$
and $R=R(d,s)$ are defined as follows,

$P=\left(\max\left\\{\frac{3d-2s}{3d-2s+2},\frac{3d-2}{3d-2+2s}\right\\},\max\left\\{\frac{3d-2s}{3d-2s+2},\frac{3d-2}{3d-2+2s}\right\\}\right),$
$Q=\left(1,\frac{d-s-1}{d-s}\right),\hskip 14.22636pt\text{and}\hskip
14.22636ptR=\left(\frac{d-s-1}{d-s},1\right).$

We have the following $L^{p}-$estimates for the operator
$\mathcal{N}_{lac}^{T}$ in dimension $d\geq 2$.

######  Theorem 4.1.

Let $d\geq 2$, $0<s\leq d$ and $0<p_{1},p_{2},p_{3}\leq\infty$ with
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}$. Then we have the following,

  1. (1)

The operator $\mathcal{N}_{lac}^{T}$ is bounded from
$L^{p_{1}}(\mathbb{R}^{d})\times L^{p_{2}}(\mathbb{R}^{d})$ to
$L^{p_{3}}(\mathbb{R}^{d})$ if

    1. (i)

$d=2$ and $(\frac{1}{p_{1}},\frac{1}{p_{2}})\in\Omega(\\{O,A,P,B\\})$,

    2. (ii)

$d\geq 3$,

       * •

$0\leq s\leq d-2$ and
$(\frac{1}{p_{1}},\frac{1}{p_{2}})\in\Omega(\\{O,A,Q,P,R,B\\})$.

       * •

$d-2\leq s\leq d$ and
$(\frac{1}{p_{1}},\frac{1}{p_{2}})\in\Omega(\\{O,A,P,B\\})$.

  2. (2)

For $d\geq 3$ and $d-2\leq s\leq d$, the operator $\mathcal{N}_{lac}^{T}$ is
bounded from $L^{p_{1}}(\mathbb{R}^{d})\times L^{p_{2}}(\mathbb{R}^{d})$ to
$L^{p_{3},\infty}(\mathbb{R}^{d})$ if $(\frac{1}{p_{1}},\frac{1}{p_{2}})$ lies
on the line segments $AQ$ and $BR$ excluding the points $Q,R$.

$O$$1/p_{1}$$1/p_{2}$$A$$B$$P$$R$$Q$ Figure 1. The figure denotes the region
of boundedness of $\mathcal{N}_{lac}^{T}$ when
$(\frac{1}{p_{1}},\frac{1}{p_{2}})$ belongs to the region
$\Omega(\\{O,A,P,B\\})$ for $d-2\leq s\leq d$ and the region
$\Omega(\\{O,A,Q,P,R,B\\})$ for $0\leq s\leq d-2$.

In dimension $d=1$, we have the following result.

######  Theorem 4.2.

Let $T$ be a compact set with finite $s-$dimensional upper Minkowski content
with $0\leq s\leq 1$. The operator $\mathcal{N}_{lac}^{T}$ is bounded from
$L^{p_{1}}(\mathbb{R})\times L^{p_{2}}(\mathbb{R})$ to $L^{p_{3}}(\mathbb{R})$
for all indices $0<p_{1},p_{2},p_{3}\leq\infty$ with
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}$ in each of the following
cases.

  1. (i)

$0\leq s\leq 1$ and
$(\frac{1}{p_{1}},\frac{1}{p_{2}})\in\Omega\left(\\{O,A,B\\}\right)$.

  2. (ii)

$0\leq s<\frac{1}{2}$ and
$(\frac{1}{p_{1}},\frac{1}{p_{2}})\in\Omega\left(\\{O,A,D,B\\}\right),$ where
$D=\left(\frac{1}{1+2s},\frac{1}{1+2s}\right).$

$B$$A$$D$$1/p$$1/q$$O$$C$ Figure 2. The figure denotes the region
$\Omega\left(\\{O,A,D,B\\}\right)$ .

##  5\. Proof of Theorem 2.4: Lacunary uncentered spherical maximal operator
$N^{T}_{lac}$

To prove Theorem 2.4, we employ a multiscale decomposition of the operator.
Let $\phi\in\mathcal{S}({\mathbb{R}}^{d})$ be a function such that
$\widehat{\phi}$ is supported in $B(0,2)$ and $\widehat{\phi}(\xi)=1$ for
$\xi\in B(0,1)$. We define $\widehat{\phi_{t}}(\xi)=\widehat{\phi}(t\xi)$ and
$\widehat{\psi}_{t}(\xi)=\widehat{\phi}(t\xi)-\widehat{\phi}(2t\xi)$. Then, we
have the identity

(5.1)
$\displaystyle\widehat{\phi}(\xi)+\sum_{j=1}^{\infty}\widehat{\psi}_{2^{-j}}(\xi)=1,\;\xi\neq
0.$

Using this identity, the lacunary maximal function $N_{lac}^{T}$ can be
dominated by

(5.2) $\displaystyle N^{T}_{lac}f(x)$ $\displaystyle\leq$
$\displaystyle\sup_{k\in\mathbb{Z}}|A^{T}_{2^{k},0}f(x)|+\sum^{\infty}_{j=1}\sup_{k\in\mathbb{Z}}|A^{T}_{2^{k},j}f(x)|,$

where

(5.3) $\displaystyle A^{T}_{2^{k},j}f(x)$ $\displaystyle=\sup_{u\in
T}|(f*\psi_{2^{k-j}}*\sigma_{2^{k}})(x+2^{k}u)|,\;j\geq 1,$ (5.4)
$\displaystyle A^{T}_{2^{k},0}f(x)$ $\displaystyle=\sup_{u\in
T}|(f*\phi_{2^{k}}*\sigma_{2^{k}})(x+2^{k}u)|.$

We need to prove suitable estimates on the intermediary lacunary operators

$M^{T}_{j}f(x)=\sup_{k\in\mathbb{Z}}|A^{T}_{2^{k},j}f(x)|,\;j\geq 0.$

First, we observe that $M_{0}^{T}f$ can be controlled by the classical Hardy-
Littlewood maximal function $M_{HL}f$.

######  Proposition 5.1.

Let $f\in L^{1}_{loc}({\mathbb{R}}^{d})$. Then

$M_{0}^{T}f(x)\lesssim M_{HL}f(x),~{}\text{a.e}~{}x\in\mathbb{R}^{d}.$

###### Proof.

Consider

$\displaystyle|A^{T}_{2^{k},0}f(x)|$ $\displaystyle=$ $\displaystyle\sup_{u\in
T}\left|\int_{\mathbb{S}^{d-1}}\phi_{2^{k}}*f(x+2^{k}(u+y))~{}d\sigma(y)\right|$
$\displaystyle\lesssim$ $\displaystyle\sup_{u\in
T,|y|=1}|\phi_{2^{k}}*f(x+2^{k}(u+y))|.$

Using the fact that $\phi\in\mathcal{S}({\mathbb{R}}^{d})$ and $T$ is compact,
we get that

$\displaystyle\left|\phi_{2^{k}}*f(x+2^{k}(u+y))\right|$ $\displaystyle=$
$\displaystyle
2^{-kd}\left|\int_{{\mathbb{R}}^{d}}f(z)\phi(2^{-k}(x-z)+u+y)~{}dz\right|$
$\displaystyle\lesssim$
$\displaystyle\left|\int_{{\mathbb{R}}^{d}}\frac{2^{-kd}f(z)}{\Big{(}1+|2^{-k}(x-z)+u+y|\Big{)}^{N}}dz\right|$
$\displaystyle\lesssim$ $\displaystyle M_{HL}f(x).$

∎

To deal with maximal operators $M_{j}^{T}$, we need a counting argument that
bounds the $L^{p}-$norm of the maximal average $A^{T}_{1,j}$ by that of
certain linear operators with appropriate growth in $j$.

######  Lemma 5.2 (Lemma 4.5,
[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres]).

Let $T$ be a compact subset of $\mathbb{R}^{d}$ and $\\{T_{j}\\}$ be the
collection of centers of balls of radius $2^{-j}$ covering $T$. Then, for
$p\geq 1,$ we have

$\|\sup_{u\in
T}|f\ast\psi_{2^{-j}}\ast\sigma(\cdot+u)|\|^{p}_{L^{p}({\mathbb{R}}^{d})}\lesssim\sharp
T_{j}\|f\ast\psi_{2^{-j}}\ast\sigma\|^{p}_{L^{p}({\mathbb{R}}^{d})}.$

We will require the following $L^{p}-$estimates for the maximal averages
$A^{T}_{1,j}$ from [NikodymSetsAndMaximalFunctionsAssociatedWithSpheres].

######  Lemma 5.3 ([NikodymSetsAndMaximalFunctionsAssociatedWithSpheres]).

Let $j\in{\mathbb{N}}$. The following bounds hold true:

(5.5) $\displaystyle\|A^{T}_{1,j}\|_{L^{1}({\mathbb{R}}^{d})\to
L^{1}({\mathbb{R}}^{d})}$ $\displaystyle\lesssim\min\\{2^{js},2^{j}\\}.$ (5.6)
$\displaystyle\|A^{T}_{1,j}\|_{L^{\frac{3}{2}}({\mathbb{R}}^{d})\to
L^{\frac{3}{2}}({\mathbb{R}}^{d})}$ $\displaystyle\lesssim
j^{\frac{1}{3}}2^{\frac{j}{6}},\;\;d\geq 3,$ (5.7)
$\displaystyle\|A^{T}_{1,j}\|_{L^{\frac{4}{3}}({\mathbb{R}}^{d})\to
L^{\frac{4}{3}}({\mathbb{R}}^{d})}$ $\displaystyle\lesssim
j^{\frac{1}{4}}2^{\frac{j}{4}},\;\;d\geq 4.$

The $L^{1}-$estimate above, is a consequence of Lemma 5.2 and the kernel
estimate
$|\psi_{2^{-j}}*d\sigma(x)|\lesssim\frac{2^{j}}{(1+2^{j}||x|-1|)^{N}}$. The
$L^{p}-$estimates at the points $p=\frac{3}{2},\frac{4}{3}$ were obtained in
[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres] by proving almost sharp
estimates for the Nikodym maximal function $N^{\delta}$ defined in Section
3.1.

Next, we illustrate a bootstrap argument which allows us to extend the range
of $p$ for which $L^{p}-$boundedness of the multiscale operator $M_{j}^{T}$
holds, provided we have an initial $L^{p}-$estimate for the single scale
operator $A_{1,j}^{T}$.

######  Lemma 5.4.

Let $1\leq p_{1}<p_{2}$ be such that

$\displaystyle\|A_{1,j}^{T}\|_{L^{p_{1}}\to L^{p_{1}}}$ $\displaystyle\leq
C_{1},$ $\displaystyle\|M_{j}^{T}\|_{L^{p_{2}}\to L^{p_{2}}}$
$\displaystyle\leq C_{2}.$

Then, we have

$\|M_{j}^{T}\|_{L^{p}\to L^{p}}\lesssim
C_{1}^{\frac{p_{1}}{2}}C_{2}^{1-\frac{p_{1}}{2}},\;\;\text{for}\;p=\frac{2p_{2}}{2+p_{2}-p_{1}}.$

###### Proof.

The proof involves a vector-value argument. Consider the following vector-
valued operator $\boldsymbol{\vec{A}}$ acting on a sequence of measurable
functions $f=\\{f_{k}\\}_{k\in{\mathbb{Z}}}$.

$\boldsymbol{\vec{A}}(\\{f_{k}\\}_{k\in{\mathbb{Z}}})(x)=\\{A_{2^{k},j}^{T}(f_{k})(x)\\}_{k\in{\mathbb{Z}}}.$

First, observe that the operator $A_{1,j}^{T}$ readily extends to a vector-
valued setting, namely we get that

(5.8) $\|\boldsymbol{\vec{A}}\|_{L^{p_{1}}(\ell_{p_{1}})\to
L^{p_{1}}(\ell_{p_{1}})}\leq C_{1}.$

Next, using the $L^{p_{2}}-$boundedness of $M_{j}^{T},$ we get that

$\displaystyle\|\boldsymbol{\vec{A}}f\|_{L^{p_{2}}(\ell_{\infty})}$
$\displaystyle=$ $\displaystyle\|\sup_{k}A^{T}_{2^{k},j}(f_{k})\|_{{p_{2}}}$
$\displaystyle\leq$ $\displaystyle\|M_{j}^{T}(\sup_{m}|f_{m}|)\|_{{p_{2}}}$
$\displaystyle\leq$ $\displaystyle C_{2}\|f\|_{L^{p_{2}}(\ell_{\infty})}.$

Therefore, we have that

(5.9) $\|\boldsymbol{\vec{A}}\|_{L^{p_{2}}(\ell_{\infty})\to
L^{p_{2}}(\ell_{\infty})}\leq C_{2}.$

Interpolate between (5.8) and (5.9) to deduce that

(5.10) $\|\boldsymbol{\vec{A}}\|_{L^{p}(\ell_{2})\to L^{p}(\ell_{2})}\lesssim
C_{1}^{\frac{p_{1}}{2}}C_{2}^{1-\frac{p_{1}}{2}}.$

This estimate can be used to get that

$\displaystyle\|M^{T}_{j}f\|_{p}$ $\displaystyle\leq$
$\displaystyle\left\|\left(\sum_{k\in\mathbb{Z}}|A^{T}_{2^{k},j}f|^{2}\right)^{\frac{1}{2}}\right\|_{p}$
$\displaystyle\lesssim$ $\displaystyle
C_{1}^{\frac{p_{1}}{2}}C_{2}^{1-\frac{p_{1}}{2}}\left\|\left(\sum_{k\in\mathbb{Z}}|\psi_{2^{k-j}}*f|^{2}\right)^{\frac{1}{2}}\right\|_{p}$
$\displaystyle\lesssim$ $\displaystyle
C_{1}^{\frac{p_{1}}{2}}C_{2}^{1-\frac{p_{1}}{2}}\|f\|_{p},$

where in the last step, we have used the Littlewood-Paley inequality (see for
example, [Duoandikoetxeabook]). ∎

We now state the main estimates for the operators $M_{j}^{T}$ needed to prove
Theorem 2.4.

######  Lemma 5.5.

Let $j\in{\mathbb{N}}$ and $0\leq s<d-1$. We have the following estimates,

  1. (1)

For $d\geq 2$,

(5.11) $\|M_{j}^{T}f\|_{2}\lesssim 2^{-\frac{(d-1-s)j}{2}}\|f\|_{2}.$

  2. (2)

For $d\geq 2,\;p_{0}=1+\frac{s}{2(d-1)}$,

(5.12) $\|M_{j}^{T}\|_{L^{p_{0}}({\mathbb{R}}^{d})\to
L^{p_{0}}({\mathbb{R}}^{d})}\lesssim 2^{j\frac{s(d-1)}{2(d-1)+s}}.$

  3. (3)

For $\frac{3}{2}<p\leq 2$,

(5.13) $\|M_{j}^{T}\|_{L^{p}({\mathbb{R}}^{3})\to
L^{p}({\mathbb{R}}^{3})}\lesssim
j^{\frac{1}{p}-\frac{1}{2}}2^{-j\left(\frac{9-4s}{2}-\frac{7-3s}{p}\right)}.$

  4. (4)

For $d\geq 4,\;\frac{4}{3}<p\leq 2$,

(5.14) $\|M_{j}^{T}\|_{L^{p}({\mathbb{R}}^{d})\to
L^{p}({\mathbb{R}}^{d})}\lesssim
j^{\frac{1}{p}-\frac{1}{2}}2^{-j\left(\frac{3d-2-3s}{2}-\frac{2d-1-2s}{p}\right)}.$

###### Proof.

Proof of (1): By an application of Lemma 5.2, we get that

$\displaystyle\|A^{T}_{1,j}f\|_{2}$ $\displaystyle=$
$\displaystyle\|\sup_{u\in T}|f\ast\psi_{2^{-j}}\ast\sigma(\cdot+u)|\|_{2}$
$\displaystyle\lesssim$ $\displaystyle
2^{\frac{js}{2}}\|f\ast\psi_{2^{-j}}\ast\sigma\|_{2}.$

We use Plancherel’s identity and standard scaling argument to get that

(5.15) $\displaystyle\|A^{T}_{2^{k},j}f\|_{2}\lesssim
2^{-\frac{(d-1-s)j}{2}}\|f\|_{2}.$

Let $\tilde{\psi}\in\mathcal{S}({\mathbb{R}}^{d})$ be such that
$\widehat{\tilde{\psi}}(\xi)=1$ on the support of $\hat{\psi}$. Since,
$\widehat{\tilde{\psi}}_{2^{k}}\widehat{{\psi}}_{2^{k}}=\widehat{{\psi}}_{2^{k}}$
for all $k,$ we can use the orthogonality of Fourier transform in the
following way to get the desired result.

$\displaystyle\|M^{T}_{j}f\|^{2}_{2}$ $\displaystyle\leq$
$\displaystyle\left\|\left(\sum_{k\in\mathbb{Z}}|A^{T}_{2^{k},j}f|^{2}\right)^{\frac{1}{2}}\right\|^{2}_{2}=\left\|\left(\sum_{k\in\mathbb{Z}}|A^{T}_{2^{k},j}(\tilde{\psi}_{2^{k-j}}*f)|^{2}\right)^{\frac{1}{2}}\right\|^{2}_{2}$
$\displaystyle\lesssim$ $\displaystyle
2^{-(d-1-s)j}\sum_{k}\|\tilde{\psi}_{2^{k-j}}*f\|^{2}_{2}$
$\displaystyle\lesssim$ $\displaystyle 2^{-(d-1-s)j}\|f\|^{2}_{2}.$

Proof of (2): By an application of Lemma 5.4 along with the estimates (5.5)
and 5.11, we have that

$\|M_{j}^{T}\|_{L^{\frac{4}{3}}\to L^{\frac{4}{3}}}\lesssim
2^{-j\frac{(d-1-3s)}{4}}.$

By a recursive application of Lemma 5.4 along with (5.5) and the above
estimate, we obtain the estimate 5.12.
Proof of (3): The inequality 5.13 follows from repeated application of Lemma
5.4 along with the estimates (5.6) and 5.11.
Proof of (4): The proof is similar to that of inequality 5.13 with the
exception of using estimate 5.7 instead of 5.6. ∎

To obtain a restricted weak type inequality, we will employ an interpolation
trick of Bourgain for $M_{j}^{T}$. We state the lemma for convenience and a
proof can be found in [Lee1] (Lemma 2.6).

######  Lemma 5.6 ([Lee1]).

Let $\epsilon_{1},\epsilon_{2}>0$. Suppose that $\\{T_{j}\\}$ is a sequence of
linear (or sublinear) operators such that for some $1\leq p_{1},p_{2}<\infty$,
and $1\leq q_{1},q_{2}<\infty$,

$\|T_{j}(f)\|_{L^{q_{1}}}\leq
M_{1}2^{\epsilon_{1}j}\|f\|_{L^{p_{1}}},~{}~{}\|T_{j}(f)\|_{L^{q_{2}}}\leq
M_{2}2^{-\epsilon_{2}j}\|f\|_{L^{p_{2}}}.$

Then $T=\sum_{j}T_{j}$ is bounded from $L^{p,1}$ to $L^{q,\infty}$, i.e.

$\|T(f)\|_{L^{q,\infty}}\lesssim
M^{\theta}_{1}M^{1-\theta}_{2}\|f\|_{L^{p,1}},$

where $\theta=\frac{\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}$,
$\frac{1}{q}=\frac{\theta}{q_{1}}+\frac{1-\theta}{q_{2}}$ and
$\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}}$.

Since we have all the ingredients, we now conclude the proof of Theorem 2.4.

###### Proof of Theorem 2.4.

First, we prove the restricted weak type inequality at the endpoint
$p_{0}=1+\frac{s}{d-1}$.

The restricted weak type inequality $N_{lac}^{T}:L^{1+\frac{s}{d-1},1}\to
L^{1+\frac{s}{d-1},\infty}$ follows from the estimates 5.11, (5.12) along with
an application of Lemma 5.6 for the operators $M_{j}^{T}$.

The proof of restricted weak type estimate for the endpoint
$p=1+\frac{1}{d-s}$ is simpler. Indeed, it follows by applying Lemma 5.6 to
the endpoint estimates 5.11 and $\|M_{j}^{T}\|_{L^{1}\to L^{1}}\lesssim
2^{j}$.

The $L^{p}-$estimates for $N^{T}_{lac}$ for the range
$p>1+\min\\{\frac{s}{d-1},\frac{1}{d-s}\\}$ follows from the interpolation of
respective restricted weak type inequalities and the trivial
$L^{\infty}-$estimate. Finally, we obtain the better $L^{p}-$bounds for the
case $p>1+\frac{5-2s}{9-4s}$ and $p>1+\frac{d-s}{3(d-s)-2}$ in dimensions
$d=3$ and $d\geq 4$ respectively by resorting to inequalities 5.13 and 5.14
for $M_{j}^{T}$ and summing in $j$. ∎

##  6\. Proof of Theorem 2.7: $L^{p}-$improving properties of $A^{T}$

In view of the real interpolation theory, we need to establish restricted
weak-type estimates for the operator $A^{T}$ at the endpoints described in
Theorem 2.7. This is obtained by decomposing the operator $A^{T}$ at dyadic
scales and a discretization of the set $T$ adapted to each dyadic scale. We
will prove suitable $L^{p}-$estimates on each piece of $A^{T}$ and use the
interpolation theorem due to Bourgain to get desired restricted weak-type
estimates.

Let $\phi$ and $\psi$ be as in 5.1. This gives us the following decomposition,

$A^{T}_{1}f(x)\leq\sum\limits_{j=0}^{\infty}A^{T}_{1,j}f(x).$

Next, consider the covering of $T$ by balls of radius $\delta=2^{-j}.$
Invoking the covering argument frm Lemma 5.2, we get that

$\|A^{T}_{1,j}|\|^{p}_{L^{p}}\lesssim
N(T,2^{-j})\|f\ast\psi_{2^{-j}}\ast\sigma\|^{p}_{L^{p}},~{}p\geq 1.$

Recall that we have the bound $N(T,2^{-j})\leq 2^{js}.$ So by using the
Fourier transform estimate

(6.1) $\displaystyle\|A^{T}_{1,j}\|_{L^{2}\to L^{2}}$ $\displaystyle\lesssim
2^{-j\left(\frac{d-s-1}{2}\right)}.$

Moreover, using the $L^{p}-$improving estimates of $A^{u}_{1}$ [Littman] (for
a fixed $u\in T$) and applying the covering argument we get

(6.2) $\|A^{T}_{1,j}\|_{L^{\frac{d+1}{d}}\to L^{d+1}}\lesssim
2^{j\left(\frac{s}{d+1}\right)}.$

Now using the estimates (6.1), (5.5), and (6.2), and applying Bourgain’s
interpolation trick [Lemma 2.6, [Lee1]] we get

$\|A^{T}_{1}f\|_{L^{q,\infty}}\lesssim\|f\|_{L^{p,1}},$

where $(1/p,1/q)=H$ and $E$. This completes the proof. ∎

##  7\. Proof of Theorems 3.1 and 3.2: Nikodym maximal operators

###  7.1. Proof of Theorem 3.1:

Theorem 3.1 follows by using similar arguments as in the case Theorem 2.4. We
give a brief sketch of the proof.

Let us first, recall the $L^{p}-$estimates for the single scale operator
$N^{\delta}_{1}$ from [Theorem 1.3,
[NikodymSetsAndMaximalFunctionsAssociatedWithSpheres]].

(7.1) $\displaystyle\|N^{\delta}_{1}\|_{L^{1}({\mathbb{R}}^{d})\to
L^{1}({\mathbb{R}}^{d})}$ $\displaystyle\lesssim\delta^{-1}$ (7.2)
$\displaystyle\|N^{\delta}_{1}\|_{L^{\frac{3}{2}}({\mathbb{R}}^{d})\to
L^{\frac{3}{2}}({\mathbb{R}}^{d})}$
$\displaystyle\lesssim\delta^{-\frac{1}{6}}(-\log\delta)^{\frac{1}{3}}$ (7.3)
$\displaystyle\|N^{\delta}_{1}\|_{L^{\frac{4}{3}}({\mathbb{R}}^{d})\to
L^{\frac{4}{3}}({\mathbb{R}}^{d})}$
$\displaystyle\lesssim\delta^{-\frac{1}{4}}(-\log\delta)^{\frac{1}{4}}.$

Denote

$N^{\delta}_{k}f(x)=\sup_{u\in\mathbb{S}^{d-1}}\frac{1}{|S^{\delta}(0)|}\Big{|}\int_{S^{\delta}(0)}f(x+2^{k}(u+y))dy\Big{|}.$

As in the proof of Theorem 3.1, using the identity (5.1), we get the following
decomposition of $N^{\delta}_{k}f.$

$N^{\delta}_{k}f(x)=N^{\delta}_{k}(\phi_{2^{k}}*f)(x)+\sum^{\infty}_{j=1}N^{\delta}_{k}(\psi_{2^{k-j}}*f)(x).$

Consequently, we get that

(7.4) $\displaystyle N^{\delta}_{lac}f(x)$ $\displaystyle\leq$
$\displaystyle\sup_{k\in\mathbb{Z}}|N^{\delta}_{k}(\phi_{2^{k}}*f)(x)|+\sum^{\infty}_{j=1}\sup_{k\in\mathbb{Z}}|N^{\delta}_{k}(\psi_{2^{k-j}}*f)(x)|.$

Let $M_{j}^{\delta}$ denote the intermediary maximal operator

$M^{\delta}_{j}f(x)=\sup_{k\in\mathbb{Z}}|N^{\delta}_{k}(\psi_{2^{k-j}}*f)(x)|.$

Observe that the arguments used in Proposition 5.1 apply here as well and we
get the corresponding estimate for $M_{0}^{\delta}f,$ namely,

$M_{0}^{\delta}f(x)\lesssim M_{HL}f(x),~{}\text{a.e}~{}x\in\mathbb{R}^{d}.$

Next, we have the following $L^{2}-$estimate for the operators
$M_{j}^{\delta}$.

######  Lemma 7.1.

For $d\geq 2$ and $j\in\mathbb{N},$ the following holds

$\|M_{j}^{\delta}f\|_{2}\lesssim\delta^{-\epsilon}2^{-j\epsilon}\|f\|_{2}.$

Consequently, we get that

$\|N^{\delta}_{lac}\|_{2}\lesssim\delta^{-\epsilon}\|f\|_{2}.$

###### Proof.

We claim that for any $0<\epsilon<1$ the following estimate holds,

(7.5)
$\frac{|\widehat{\chi_{S^{\delta}(0)}}(\xi)|}{|S^{\delta}(0)|}\lesssim\frac{\delta^{-\epsilon}}{(1+|\xi|)^{\frac{d-1}{2}+\epsilon}}.$

Indeed, to prove the above inequality it is enough to show

$|\widehat{\chi_{S^{\delta}(0)}}(\xi)|\lesssim\min\left\\{\frac{1}{(1+|\xi|)^{\frac{d+1}{2}}},\frac{\delta}{(1+|\xi|)^{\frac{d-1}{2}}}\right\\}.$

These estimates can be obtained by using the following identity and mean value
theorem,

$\displaystyle\widehat{\chi_{S^{\delta}(0)}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{(1+\delta)^{\frac{d}{2}}J_{\frac{d}{2}}((1+\delta)|\xi|)}{|\xi|^{\frac{d}{2}}}-\frac{(1-\delta)^{\frac{d}{2}}J_{\frac{d}{2}}((1-\delta)|\xi|)}{|\xi|^{\frac{d}{2}}}.$

From the covering argument of Lemma 5.2 and (7.5), we obtain that

(7.6) $\displaystyle\|N^{\delta}_{1}(\psi_{2^{-j}}*f)\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\sup_{u\in\mathbb{S}^{d-1}}\left|f\ast\psi_{2^{-j}}\ast\frac{\chi_{S^{\delta}(0)}}{|S^{\delta}(0)|}(\cdot+u)\right|\right\|_{2}$
$\displaystyle\lesssim$ $\displaystyle
2^{j\frac{d-1}{2}}\left\|f\ast\psi_{2^{-j}}\ast\frac{\chi_{S^{\delta}(0)}}{|S^{\delta}(0)|}(\cdot)\right\|_{2}$
$\displaystyle\lesssim$ $\displaystyle
2^{j\frac{d-1}{2}}\frac{\delta^{-\epsilon}}{2^{j\left(\frac{d-1}{2}+\epsilon\right)}}\|f\|_{2}$
$\displaystyle=$ $\displaystyle\delta^{-\epsilon}2^{-j\epsilon}\|f\|_{2}.$

By an scaling argument and the estimate (7.6), we have that

$\displaystyle\|M^{\delta}_{j}f\|^{2}_{2}$ $\displaystyle\leq$
$\displaystyle\left\|\left(\sum_{k\in\mathbb{Z}}|N^{\delta}_{k}(\psi_{2^{k-j}}*f)|^{2}\right)^{\frac{1}{2}}\right\|^{2}_{2}=\left\|\left(\sum_{k\in\mathbb{Z}}|N^{\delta}_{k}(\psi_{2^{k-j}}*\widetilde{\psi}_{2^{k-j}}*f)|^{2}\right)^{\frac{1}{2}}\right\|^{2}_{2}$
$\displaystyle\lesssim$
$\displaystyle\delta^{-2\epsilon}2^{-2j\epsilon}\sum_{k}\|\widetilde{\psi}_{2^{k-j}}*f\|^{2}_{2}\lesssim\delta^{-2\epsilon}2^{-2j\epsilon}\|f\|^{2}_{2}.$

Here, in the last step we have used the Littlewood-Paley inequality. ∎

By a bootstrapping argument similar to that in the proof of Theorem 2.4 with
estimates (7.1), (7.2), (7.3) and Lemma 7.1, we obtain the required estimates
for $N^{\delta}_{lac}$.

A recursive application of the arguments similar to that of Lemma 5.4 for
$M^{\delta}_{j}$ along with Lemma 7.1 and the estimates (7.2), (7.3) gives us
that

(7.7) $\displaystyle\|M_{j}^{\delta}\|_{L^{p}\to L^{p}}$
$\displaystyle\lesssim
2^{-j\epsilon\left(4-\frac{6}{p}\right)}\delta^{-\epsilon}\delta^{\frac{1}{p}-\frac{1}{2}},$
$\displaystyle\text{for}\;d\geq 3,\;\frac{3}{2}<p\leq 2,$ (7.8)
$\displaystyle\|M_{j}^{\delta}\|_{L^{p}\to L^{p}}$ $\displaystyle\lesssim
2^{-j\epsilon\left(3-\frac{4}{p}\right)}\delta^{-\epsilon}\delta^{\frac{1}{p}-\frac{1}{2}},$
$\displaystyle\text{for}\;d\geq 4,\;\frac{4}{3}<p\leq 2.$

∎

###  7.2. Proof of Theorem 3.2:

The following result will be useful in computing the $L^{\frac{1}{2}}-$norm of
the operator using appropriate $L^{1}-$bounds.

######  Lemma 7.2 ([IPS]).

Let $T$ be a bilinear operator satisfying the following.

  1. (i)

There exists $N>0$ such that for functions $f_{i},f_{j}$ supported in the unit
cubes $Q_{l_{i}}$ and $Q_{l_{j}}$ with their lower left corners at $l_{i}$ and
$l_{j}$ respectively and $\|l_{i}-l_{j}\|_{\infty}>N,$ we have
$T(f_{i},f_{j})=0.$

  2. (ii)

There exists $R>0$ such that $T(f,g)$ is supported on
$(\mathrm{supp}(f)+B(0,R))\bigcup(\mathrm{supp}(g)+B(0,R)).$

  3. (iii)

The operator $T$ satisfies the $L^{p}-$estimate

$\|T(f,g)\|_{p_{3}}\leq C\|f\|_{p_{1}}\|g\|_{p_{2}},$

for all functions $f,g$ supported in a fixed cube and for some exponents
$p_{1},p_{2}$ and $p_{3}$ with $p_{1},p_{2}\geq
1,\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}>1$ and $p_{3}\geq p$.

Then, for all $r\in[p,p_{3}],$ we have

$\|T\|_{L^{p_{1}}\times L^{p_{2}}\to L^{r}}\lesssim C.$

Now we proceed with the proof of Theorem 3.2. First, we prove the boundedness
of $\mathcal{N}^{\delta}$ in the Banach range. We observe the trivial
$L^{\infty}$ estimate

(7.9) $\displaystyle\|\mathcal{N}^{\delta}\|_{L^{\infty}\times L^{\infty}\to
L^{\infty}}\leq 1.$

The Hölder’s inequality yields

$\mathcal{N}^{\delta}(f,g)(x)\leq N_{p}f(x)N_{p^{\prime}}g(x),\;\;\text{for
}\frac{1}{p}+\frac{1}{p^{\prime}}=1,\;p\geq 1,$

where
$N_{p}f(x)=\sup\limits_{u\in\mathbb{S}^{d-1}}\left(\frac{1}{\delta}\int_{S^{\delta}(0)}|f(x+u+y)|^{p}\;d(y,z)\right)^{\frac{1}{p}}.$

We use a change of variable to polar co-ordinates and the well-known slicing
technique, see
[MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators],
to prove the following estimates for $N_{p}f.$ Consider

$\displaystyle\|N_{p}f\|_{p}^{p}$
$\displaystyle\leq\int_{{\mathbb{R}}^{d}}\sup\limits_{u\in\mathbb{S}^{d-1}}\frac{1}{\delta}\int_{t=1-\delta}^{1+\delta}\int_{\mathbb{S}^{2d-1}}|f(x+u+ty)|^{p}\;d\sigma_{2d}(y,z)t^{2d-1}dtdx$
$\displaystyle\lesssim\int_{{\mathbb{R}}^{d}}\sup\limits_{u\in\mathbb{S}^{d-1}}\frac{1}{\delta}\int_{t=1-\delta}^{1+\delta}\int_{B_{d}(0,1)}|f(x+u+ty)|^{p}\;(1-|y|^{2})^{\frac{d-2}{2}}dydtdx$
$\displaystyle\lesssim\int_{{\mathbb{R}}^{d}}\sup\limits_{u\in\mathbb{S}^{d-1}}\frac{1}{\delta}\int_{t=1-\delta}^{1+\delta}\int_{B_{d}(u,t)}|f(x+y)|^{p}\;dyt^{-d}dtdx$
$\displaystyle\lesssim\int_{{\mathbb{R}}^{d}}\frac{1}{\delta}\int_{t=1-\delta}^{1+\delta}\int_{B_{d}(0,10)}|f(x+y)|^{p}\;dydtdx$
$\displaystyle\leq\frac{1}{\delta}\int_{t=1-\delta}^{1+\delta}\int_{B_{d}(0,10)}\|f\|_{p}^{p}\;dydt$
$\displaystyle\lesssim\|f\|_{p}^{p}$

The estimate above implies that
$\|\mathcal{N}^{\delta}(f,g)\|_{1}\lesssim\|f\|_{p}\|g\|_{p^{\prime}}.$ The
desired result for other exponents in the Banach triangle follows by
interpolation.

Next, we prove the $L^{1}\times L^{1}\to L^{\frac{1}{2}}$ bound of
$\mathcal{N}^{\delta}$. We can see that

$\displaystyle\mathcal{N}^{\delta}(f,g)(x)$
$\displaystyle\leq\frac{1}{\delta}\left|\int_{B_{2d}(0,10)}f(x+y)g(x+z)\;d(y,z)\right|$
$\displaystyle\leq\frac{1}{\delta}\int_{B_{d}(0,10)}|f(x+y)|\;dy\int_{B_{d}(0,10)}|g(x+z)|\;dz$
$\displaystyle\leq\frac{1}{\delta}\|f\|_{1}\int_{B_{d}(0,10)}|g(x+z)|\;dz.$

From the above estimates, we obtain

$\|\mathcal{N}^{\delta}(f,g)\|_{1}\lesssim\frac{1}{\delta}\|f\|_{1}\|g\|_{1}.$

The required $L^{1}(\mathbb{R}^{d})\times L^{1}(\mathbb{R}^{d})\to
L^{\frac{1}{2}}(\mathbb{R}^{d})-$estimate follows from an application of Lemma
7.2. ∎

##  8\. Proof of Theorem 4.1: Lacunary bilinear maximal operator
$\mathcal{N}_{lac}^{T}$ in dimension $d\geq 2$

In order to prove this theorem, we decompose the operator
$\mathcal{N}^{T}_{lac}$ into infinitely many pieces. Indeed, using the
identity (5.1) we get for any given $k\in\mathbb{Z}$,

$\mathcal{N}^{T}_{2^{k}}(f,g)(x)\leq\mathcal{A}^{\phi,\phi}_{2^{k}}(f,g)(x)+\mathcal{A}^{\phi,\infty}_{2^{k}}(f,g)(x)+\mathcal{A}^{\infty,\phi}_{2^{k}}(f,g)(x)+\sum_{i,j\geq
1}\mathcal{A}^{i,j}_{2^{k}}(f,g)(x),$

where

$\displaystyle\mathcal{A}^{\phi,\phi}_{2^{k}}(f,g)(x)=\mathcal{N}^{T}_{2^{k}}(f*\phi_{2^{k}},g*\phi_{2^{k}})(x),~{}~{}$
$\displaystyle\mathcal{A}^{\phi,\infty}_{2^{k}}(f,g)(x)=\mathcal{N}^{T}_{2^{k}}(f*\phi_{2^{k}},g)(x),$
$\displaystyle\mathcal{A}^{\infty,\phi}_{2^{k}}(f,g)(x)=\mathcal{N}^{T}_{2^{k}}(f,g*\phi_{2^{k}})(x),~{}~{}\text{and}~{}~{}$
$\displaystyle\mathcal{A}_{2^{k}}^{i,j}(f,g)(x)=\mathcal{N}_{2^{k}}^{T}(f*\psi_{2^{k-i}},g*\psi_{2^{k-j}})(x).$

Observe that

$\displaystyle\mathcal{N}^{T}_{2^{k}}(f*\phi_{2^{k}},g)(x)$
$\displaystyle\leq$ $\displaystyle\sup_{u\in T_{1},|y|\leq
1}|\phi_{2^{k}}*f(x+2^{k}(u+y))|\sup_{v\in
T_{2}}\int_{\mathbb{S}^{2d-1}}|g(x+2^{k}(v+z))|d\sigma(y,z)$ $\displaystyle=$
$\displaystyle\sup_{u\in T_{1},|y|\leq
1}|\phi_{2^{k}}*f(x+2^{k}(u+y))|\sup_{v\in
T_{2}}\int_{B^{d}(0,1)}|g(x+2^{k}(v+z))|(1-|z|^{2})^{\frac{d-2}{2}}\int_{\mathbb{S}^{d-1}}d\sigma(y)dz.$

Now, using Proposition 5.1 and boundedness of Hardy–Littlewood maximal
function $M_{HL}$ we get $L^{1}(\mathbb{R}^{d})\times
L^{1}(\mathbb{R}^{d})\rightarrow
L^{\frac{1}{2},\infty}(\mathbb{R}^{d})-$estimate of the maximal function
$\sup_{k\in\mathbb{Z}}\mathcal{N}^{T}_{2^{k}}(f*\phi_{2^{k}},g)$. The similar
estimate holds for $\sup_{k\in\mathbb{Z}}\mathcal{A}^{\phi,\phi}_{2^{k}}(f,g)$
and $\sup_{k\in\mathbb{Z}}\mathcal{A}^{\infty,\phi}_{2^{k}}(f,g)$. Therefore,
it remains to study boundedness of $\mathcal{A}_{2^{k}}^{i,j}(f,g)$.

We prove the following $L^{p}-$estimates for the operator
$\mathcal{A}_{1}^{i,j}.$

######  Lemma 8.1.

Let $d\geq 1$ and $0<s\leq d$. Then, for $i,j\geq 0,$ we have

  1. (i)

$\|\mathcal{A}_{1}^{i,j}\|_{L^{2}\times L^{2}\to L^{1}}\lesssim
2^{-(i+j)(\frac{3d-2-2s}{4})},$

  2. (ii)

$\|\mathcal{A}_{1}^{i,j}\|_{L^{1}\times L^{1}\to
L^{\frac{1}{2}}}\lesssim\min\\{2^{(i+j)},2^{(i+j)s}\\}.$

Consequently, for $1\leq p\leq 2,$ we get that,

(8.1) $\displaystyle\|\mathcal{A}_{1}^{i,j}\|_{L^{p}\times L^{p}\to
L^{p/2}}\lesssim
2^{-(i+j)\max\left\\{\left(\frac{3d-2}{2}-\frac{3d-2+2s}{2p}\right),\left(\frac{3d-2s}{2}-\frac{3d+2-2s}{2p}\right)\right\\}}.$

###### Proof.

Observe that the operator $\mathcal{A}_{1}^{i,j}$ is local in nature, i.e. if
$f$ and $g$ are supported in the unit cubes, then the support of
$\mathcal{A}_{1}^{i,j}(f,g)$ lies in a fixed neighbourhood of the unit ball.
Thus, in view of Lemma 7.2, it suffices to prove
$L^{2}({\mathbb{R}}^{d})\times L^{2}({\mathbb{R}}^{d})\rightarrow
L^{2}({\mathbb{R}}^{d})-$estimates for the operator $\mathcal{A}_{1}^{i,j}$
acting on functions supported in a fixed cube. In this direction, the
following estimate for bilinear averages corresponding to fixed parameters
$u\in T_{1},v\in T_{2}$ is known, see [Proposition $4.1$, [GGPP]].

(8.2)
$\|\mathcal{A}_{u,v}^{i,j}(f,g)\|_{2}\lesssim(2^{i}+2^{j})^{-\frac{2d-1}{2}}2^{\min\\{i,j\\}\frac{d}{2}}\|f\|_{2}\|g\|_{2}.$

Next, we use the covering argument to push the estimate above for the operator
$\mathcal{A}_{1}^{i,j}$. Let $C_{k}$ denote the set of centers of balls of
radius $2^{-i}$ covering $T_{k},~{}k=1,2.$ Set
$\Psi_{2^{-j}}(x)=2^{jn}(1+2^{j}|x|)^{-(n+1)}$ and observe that
$|\tilde{\psi}_{2^{-j}}(x)|\leq\Psi_{2^{-j}}(x).$ Consider

$\displaystyle\|\mathcal{A}^{i,j}_{1}(f,g)\|^{2}_{2}$
$\displaystyle=\int\sup_{u,v}|(f\otimes
g)\ast(\psi_{2^{-i}}\otimes\psi_{2^{-j}})\ast\sigma_{2d-1}\ast(\tilde{\psi}_{2^{-i}}\otimes\tilde{\psi}_{2^{-j}})(x+u,x+v)|^{2}\;dx$
$\displaystyle\leq\int\sup_{u,v}\left|\int\int|(f\otimes
g)\ast(\psi_{2^{-i}}\otimes\psi_{2^{-j}})\ast\sigma_{2d-1}(x+u-y,x+v-z)|\Psi_{2^{-i}}\otimes\Psi_{2^{-j}}(y,z)\;dydz\right|^{2}\;dx$
$\displaystyle\leq\sum\limits_{u\in C_{1},v\in C_{2}}\int\int\int|(f\otimes
g)\ast(\psi_{2^{-i}}\otimes\psi_{2^{-j}})\ast\sigma_{2d-1}|^{2}(x+u-y,x+v-z)\Psi_{2^{-i}}\otimes\Psi_{2^{-j}}(y,z)\;dydzdx$
$\displaystyle\lesssim\sum\limits_{u\in C_{1},v\in C_{2}}\|(f\otimes
g)\ast(\psi_{2^{-i}}\otimes\psi_{2^{-j}})\ast\sigma_{2d-1}(\cdot+u,\cdot+v)\|^{2}_{2}$
$\displaystyle\lesssim\sum\limits_{u\in C_{1},v\in
C_{2}}(2^{i}+2^{j})^{-(2d-1)}2^{\min\\{i,j\\}{d}}\|f\|^{2}_{2}\|g\|^{2}_{2}$
$\displaystyle\lesssim(2^{i}+2^{j})^{-(2d-1)+s}2^{\min\\{i,j\\}{d}}\|f\|^{2}_{2}\|g\|^{2}_{2}.$

Note that here we have used the estimate (8.2) in the second to the last step.
The estimate above along with the Lemma 7.2 yields the desired estimate ((i))
in Lemma 8.1.

Next, estimate ((ii)) follows by using essentially the similar idea as
earlier. Invoking [Proposition 4.2, [IPS]] we have

$\|\mathcal{A}_{u,v}^{i,j}(f,g)\|_{1}\lesssim\|f\|_{1}\|g\|_{1}.$

Observe that using the slicing argument, we get that

$\displaystyle\mathcal{A}_{1}^{i,j}(f,g)(x)$ $\displaystyle=\sup_{(u,v)\in
T}\bigg{|}\int_{0}^{1}r^{d-1}(1-r^{2})^{\frac{d-2}{2}}\int_{\mathbb{S}^{d-1}}f*\psi_{2^{-i}}(x+u+ry)\;d\sigma(y)$
$\displaystyle\hskip
56.9055pt\int_{\mathbb{S}^{d-1}}g*\psi_{2^{-j}}(x+v+\sqrt{1-r^{2}}z)\;d\sigma(z)\;dr\bigg{|}$
$\displaystyle\leq\int_{0}^{1}r^{d-1}(1-r^{2})^{\frac{d-2}{2}}\sup_{u\in
T_{1}}\left|\int_{\mathbb{S}^{d-1}}f*\psi_{2^{-i}}(x+u+ry)\;d\sigma(y)\right|$
$\displaystyle\hskip 56.9055pt\sup_{v\in
T_{2}}\left|\int_{\mathbb{S}^{d-1}}g*\psi_{2^{-j}}(x+v+\sqrt{1-r^{2}}z)\;d\sigma(z)\right|\;dr.$

It is straightforward to verify that the kernel estimate
$|\psi_{2^{-i}}*d\sigma_{r}(x+u)|\lesssim\frac{2^{i}}{r^{d-1}(1+|x|)^{N}}$
holds with implicit constant independent of $u.$ This implies that

$\displaystyle\|\mathcal{A}_{1}^{i,j}(f,g)\|_{1}$ $\displaystyle\lesssim
2^{i+j}\int_{0}^{1}\frac{dr}{\sqrt{1-r^{2}}}\|f\|_{1}\|g\|_{1}\lesssim
2^{i+j}\|f\|_{1}\|g\|_{1}.$

As earlier, the estimate above with Lemma 7.2 yields the desired estimate
((ii)). ∎

Now, we will prove Theorem 4.1.
Proof of boundedness in the region $\Omega(\\{O,A,P,B\\})$: By interpolation,
it suffices to prove weak-type estimates at the points $A,B$ and strong type
bounds on the line segment $OP$ (excluding $P$). The weak-type bounds at the
points $A$ and $B$ follow by the following pointwise inequality

$\mathcal{N}^{T}_{lac}(f,g)(x)\lesssim\min\\{\|f\|_{\infty}M_{HL}g(x),\|g\|_{\infty}M_{HL}f(x)\\}.$

Indeed, for $d\geq 2,$ the estimate above follows using a slicing argument. We
have,

$\displaystyle\mathcal{N}^{T}_{lac}(f,g)(x)$
$\displaystyle=2\sup_{k\in\mathbb{Z}}\sup_{(u,v)\in
T}\Big{|}\int_{B^{d}(0,1)}f(x+2^{k}(u+y))(1-|y|^{2})^{\frac{d-2}{2}}\int_{\mathbb{S}^{d-1}}g(x+2^{k}(v+\sqrt{1-|y|^{2}}z))d\sigma(z)dy\Big{|}$
$\displaystyle\lesssim M_{HL}f(x)\|g\|_{L^{\infty}}.$

The other estimate follows similarly.

Next, we need to prove $L^{p}-$boundedness at points on the line segment $OP$
(excluding $P$). Fix $p>1+\min\\{\frac{2s}{3d-2},\frac{2}{3d-2s}\\}$. Using
standard dilation argument, we have that
$\mathcal{A}^{i,j}_{2^{k}}(f,g)(x)=\mathcal{A}^{i,j}_{1}(D_{2^{k}}f,D_{2^{k}}g)(2^{-k}x),$
where $D_{2^{k}}f(x)=f(2^{k}x).$ Therefore,

(8.3) $\displaystyle\|\mathcal{A}_{2^{k}}^{i,j}\|_{L^{p}\times L^{p}\to
L^{\frac{p}{2}}}=\|\mathcal{A}_{1}^{i,j}\|_{L^{p}\times L^{p}\to
L^{\frac{p}{2}}}=:C(i,j,p),$

where the operator norm satisfies the following bound (see Lemma 8.1),

$C(i,j,p)\leq
2^{-(i+j)\max\left\\{\left(\frac{3d-2}{2}-\frac{3d-2+2s}{2p}\right),\left(\frac{3d-2s}{2}-\frac{3d+2-2s}{2p}\right)\right\\}}.$

Since $\sup\limits_{|k|\leq
L}\mathcal{N}^{T}_{2^{k}}(f,g)(x)\uparrow\mathcal{N}^{T}_{lac}(f,g)(x)$ as
$L\to\infty$ in view of the monotone convergence theorem, it is enough to
prove that

(8.4) $\displaystyle\left\|\sup_{|k|\leq
L}\mathcal{N}^{T}_{2^{k}}(f,g)\right\|_{L^{\frac{p}{2}}}\leq\alpha_{L}(p)\|f\|_{L^{p}}\|g\|_{L^{p}},$

where the constant $\alpha_{L}(p)$ is independent of $L.$

We consider the following vector-valued operator to complete the proof, also
see [ImprovedBoundsForTheBilinearSphericalMaximalOperators] for a similar
argument.

$\mathfrak{M}^{i,j}_{L}:\big{(}\\{f_{k}:|k|\leq L\\},\\{g_{k}:|k|\leq
L\\}\big{)}\mapsto\Big{\\{}\mathcal{A}_{2^{k}}^{i,j}(f_{k},g_{k}):|k|\leq
L\Big{\\}}$

Now, invoking the estimate (8.4) we get

(8.5) $\displaystyle\left\|\mathfrak{M}^{i,j}_{L}\big{(}\\{f_{k}\\}_{|k|\leq
L},\\{g_{k}\\}_{|k|\leq L}\big{)}\right\|_{L^{\frac{p}{2}}(\ell_{\infty})}$
$\displaystyle=$ $\displaystyle\left\|\sup_{|k|\leq
L}\mathcal{A}_{2^{k}}^{i,j}(f_{k},g_{k})\right\|_{L^{\frac{p}{2}}}$
$\displaystyle\lesssim$ $\displaystyle\left\|\sup_{|k|\leq
L}\mathcal{A}_{2^{k}}\Big{(}M_{HL}(\sup_{|k|\leq
L}|f_{k}|),M_{HL}(\sup_{|k|\leq L}|g_{k}|)\Big{)}\right\|_{L^{\frac{p}{2}}}$
$\displaystyle\leq$ $\displaystyle\alpha_{L}(p)\left\|M_{HL}(\sup_{|k|\leq
L}|f_{k}|)\right\|_{L^{p}}\left\|M_{HL}(\sup_{|k|\leq
L}|g_{k}|)\right\|_{L^{p}}$ $\displaystyle\lesssim$
$\displaystyle\alpha_{L}(p)\left\|\\{f_{k}\\}_{|k|\leq
L}\|_{L^{p}(\ell_{\infty})}\right\|\\{g_{k}\\}_{|k|\leq
L}\|_{L^{p}(\ell_{\infty})}.$

On the other hand, using (8.3) we get

(8.6) $\displaystyle\left\|\mathfrak{M}^{i,j}_{L}\big{(}\\{f_{k}\\}_{|k|\leq
L},\\{g_{k}\\}_{|k|\leq
L}\big{)}\right\|_{L^{\frac{p}{2}}(\ell_{\frac{p}{2}})}$ $\displaystyle=$
$\displaystyle\left\|\Big{\\{}\mathcal{A}_{2^{k}}^{i,j}(f_{k},g_{k}):|k|\leq
L\Big{\\}}\right\|_{L^{\frac{p}{2}}(\ell_{\frac{p}{2}})}$ $\displaystyle\leq$
$\displaystyle C(i,j,p)\Big{(}\sum_{|k|\leq
L}\|f_{k}\|^{\frac{p}{2}}_{L^{p}}\|g_{k}\|^{\frac{p}{2}}_{L^{p}}\Big{)}^{\frac{2}{p}}$
$\displaystyle\leq$ $\displaystyle C(i,j,p)\left\|\\{f_{k}\\}_{|k|\leq
L}\right\|_{L^{p}(\ell_{p})}\left\|\\{g_{k}\\}_{|k|\leq
L}\right\|_{L^{p}(\ell_{p})}$ $\displaystyle\leq$ $\displaystyle
C(i,j,p)\left\|\\{f_{k}\\}_{|k|\leq
L}\right\|_{L^{p}(\ell_{1})}\left\|\\{g_{k}\\}_{|k|\leq
L}\right\|_{L^{p}(\ell_{1})}.$

Interpolate between estimates (8.5) and (8.6) to get that

$\displaystyle\left\|\mathfrak{M}^{i,j}_{L}\Big{(}\\{f_{k}\\}_{|k|\leq
L}\times\\{g_{k}\\}_{|k|\leq
L}\Big{)}\right\|_{L^{\frac{p}{2}}(\ell_{p})}\leq(\alpha_{L}(p)C(i,j,p))^{\frac{1}{2}}\left\|\\{f_{k}\\}_{|k|\leq
L}\right\|_{L^{p}(\ell_{2})}\left\|\\{g_{k}\\}_{|k|\leq
L}\right\|_{L^{p}(\ell_{2})}.$

Applying Littlewood–Paley theory, we get

$\left\|\sup_{|k|\leq
L}\mathcal{A}^{i,j}_{2^{k}}(f,g)\right\|_{L^{\frac{p}{2}}}\leq\alpha_{L}(p)^{1/2}C(i,j,p)^{1/2}\|f\|_{L^{p}}\|g\|_{L^{p}}.$

Therefore, when $p\geq 2$, we obtain

$\displaystyle\alpha_{L}(p)$ $\displaystyle\leq$
$\displaystyle\alpha_{L}(p)^{1/2}\sum_{i,j\geq 1}C(i,j,p)^{1/2}+1$
$\displaystyle\implies~{}~{}\alpha_{L}(p)$ $\displaystyle\lesssim$
$\displaystyle\Big{(}\sum_{i,j\geq 1}C(i,j,p)^{1/2}\Big{)}^{2}+1<\infty.$

When $1+\min\\{\frac{2s}{3d-2},\frac{2}{3d-2s}\\}<p<2$, we have

$\alpha_{L}(p)\lesssim\Big{(}\sum_{i,j\geq
1}C(i,j,p)^{\frac{p}{2}}\Big{)}^{\frac{2}{p}}+1<\infty.$

Proof of boundedness in the region $\Omega(\\{O,A,Q,P,R,B\\})$: It remains to
prove the boundedness in the triangles $\Omega(\\{A,Q,P\\})$ and
$\Omega(\\{P,R,B\\})$. We prove the boundedness in the region
$\Omega(\\{A,Q,P\\})$ and the other follows by symmetry. Moreover, by
interpolation, it is enough to prove weak type bounds on the line segment
$AQ$, excluding the point $Q$.

Let $A_{l}=\\{y\in{\mathbb{R}}^{d}:2^{-l}<\sqrt{1-|y|^{2}}\leq 2^{-l+1}\\}$.
By slicing argument, we can write the integral as

$\displaystyle\left|\int_{\mathbb{S}^{2d-1}}f(x+2^{k}(u+y))g(x+2^{k}(v+z))\;d\sigma_{2d-1}(y,z)\right|$
$\displaystyle=$ $\displaystyle
2\left|\int_{B(0,1)}f(x+2^{k}(u+y))(1-|y|^{2})^{\frac{d-2}{2}}\int_{\mathbb{S}^{d-1}}g(x+2^{k}(v+\sqrt{1-|y|^{2}}z))\;d\sigma(z)dy\right|$
$\displaystyle=$ $\displaystyle
2\left|\sum_{l=1}^{\infty}\int_{A_{l}}f(x+2^{k}(u+y))(1-|y|^{2})^{\frac{d-2}{2}}\int_{S^{d-1}}g(x+2^{k}(v+\sqrt{1-|y|^{2}}z))\;d\sigma(z)dy\right|$
$\displaystyle\lesssim$
$\displaystyle\sum_{l=1}^{\infty}2^{-l(d-2)}\left(\sup_{2^{-l}<r\leq
2^{-l+1}}\left|\int_{\mathbb{S}^{d-1}}g(x+2^{k}(v+rz))\;d\sigma(z)\right|\right)\int_{A_{l}}|f(x+2^{k}(u+y))|~{}dy.$

Therefore, we get that

$\displaystyle\mathcal{N}_{lac}^{T}(f,g)(x)\lesssim\sum_{l=1}^{\infty}2^{-l(d-2)}N_{T,l}(g)(x)M_{l}(f)(x),$

where

$\displaystyle N_{T,l}(g)(x)$ $\displaystyle=$
$\displaystyle\sup_{k\in{\mathbb{Z}}}\sup_{v\in T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}\left|\int_{\mathbb{S}^{d-1}}g(x+2^{k}(v+rz))\;d\sigma(z)\right|$
$\displaystyle=$ $\displaystyle\sup_{k\in{\mathbb{Z}}}\sup_{v\in
T_{2}}\sup_{2^{-l}<r\leq 2^{-l+1}}|g\ast\sigma_{2^{k}r}(x+2^{k}v)|$

and

$M_{l}(f)(x)=\sup_{k\in{\mathbb{Z}}}\sup_{u\in
T_{1}}\int_{A_{l}}|f(x+2^{k}(u+y))|~{}dy.$

We can see that $M_{l}$ maps $L^{1}(\mathbb{R}^{d})$ to
$L^{1,\infty}(\mathbb{R}^{d})$ for all $1\leq l<\infty$ with constant
independent of $l$. Now, we provide $L^{p}-$ estimates for the intermediary
operators $N_{T,l}$.

######  Lemma 8.2.

Let $d\geq 3$ and $0\leq s<d-2$. Suppose that $T\subset{\mathbb{R}}^{d}$ is a
compact set with finite upper Minkowski content $s$. Then

$\|N_{T,l}(f)\|_{p}\lesssim
2^{l(\frac{2d-s}{p}-d+s)}\|f\|_{p}\quad\text{for}\quad p>1+\frac{1}{d-s-1}.$

###### Proof.

We fix $2^{-l}<r\leq 2^{-l+1}$. Then we have the following decomposition of
the measure $\sigma_{2^{k}r}$ by identity 5.1,

$\sigma_{2^{k}r}=\phi_{2^{k+l}r}\ast\sigma_{2^{k}r}+\sum_{j=1}^{l}\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}+\sum_{j=1}^{\infty}\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}.$

Therefore, we have

$\displaystyle N_{T,l}(f)(x)$ $\displaystyle\leq$
$\displaystyle\sup_{k\in{\mathbb{Z}}}\sup_{v\in T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\phi_{2^{k+l}r}\ast\sigma_{2^{k}r}(x+2^{k}v)|$
$\displaystyle+\sum_{j=1}^{l}\sup_{k\in{\mathbb{Z}}}\sup_{v\in
T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(x+2^{k}v)|$
$\displaystyle+\sum_{j=1}^{\infty}\sup_{k\in{\mathbb{Z}}}\sup_{v\in
T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}(x+2^{k}v)|$
$\displaystyle=:$ $\displaystyle I+II+III.$

For the first term, we have the following kernel estimate

$|\phi_{2^{k+l}r}\ast\sigma_{2^{k}r}(x+2^{k}v)|\lesssim
2^{-kd}\left(1+\frac{|x+2^{k}v|}{2^{k}}\right)^{-N}\lesssim
2^{-kd}(1+2^{-k}|x|)^{-N}.$

This implies that $I\lesssim M_{HL}f(x)$ and hence, we have
$L^{p}-$boundedness of the term $I$ for all $1<p\leq\infty.$

The kernel estimate of the second term

$|\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(x+2^{k}v)|\lesssim
2^{(j-k)d}\left(1+\frac{|x+2^{k}v|}{2^{k}}\right)^{-N}\lesssim
2^{(j-k)d}(1+2^{-k}|x|)^{-N},$

gives that

(8.7) $\sup_{k\in{\mathbb{Z}}}\sup_{v\in T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(x+2^{k}v)|\lesssim
2^{jd}M_{HL}f(x).$

By the covering argument similar to Lemma 5.2, we can get that

$\displaystyle\left\|\sup_{v\in T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{2}$
$\displaystyle\lesssim$ $\displaystyle
2^{\frac{js}{2}}\left\|\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(\cdot)|\right\|_{2}$
$\displaystyle\lesssim$ $\displaystyle 2^{\frac{js-(j-l)(d-2)}{2}}\|f\|_{2},$

where the last inequality can be obtained from the method for
$L^{2}-$boundedness of the spherical maximal function $(d\geq
3)$[SteinMaximalFunctionsSpherical]. From the Littlewood-Paley inequality, we
can get that

$\displaystyle\left\|\sup_{k\in{\mathbb{Z}}}\sup_{v\in
T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{2}$
$\displaystyle\lesssim\left(\sum_{k\in{\mathbb{Z}}}\left\|\sup_{v\in
T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{2}^{2}\right)^{\frac{1}{2}}$
(8.8) $\displaystyle\lesssim 2^{\frac{js-(j-l)(d-2)}{2}}\|f\|_{2}.$

Interpolating between the $L^{1}-$estimate (8.7) and $L^{2}-$estimate (8.8),
we get that

$\left\|\sup_{k\in{\mathbb{Z}}}\sup_{v\in T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k+l-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{p}\lesssim
2^{j(\frac{3d-2-s}{p}-2d+2+s)}2^{l(d-2-\frac{d-2}{p})}\|f\|_{p}.$

Now summing in $j$, we get that

$\left\|II\right\|_{p}\lesssim 2^{l(\frac{2d-s}{p}-d+s)}\|f\|_{p}.$

For the third term, we have the kernel estimate

$\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}(x+2^{k}v)\lesssim\frac{2^{j}}{2^{kd}r^{d}}\left(1+\frac{|x+2^{k}v|}{2^{k}r}\right)^{-N}\lesssim\frac{2^{j}}{r^{d}}\chi_{B(0,2)(x)}+\frac{2^{j}}{(1+|x|)^{N}}\chi_{B(0,2)^{c}}(x).$

Thus, we have

(8.9) $\sup_{k\in{\mathbb{Z}}}\sup_{v\in T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}(x+2^{k}v)|\lesssim
2^{j}2^{ld}M_{HL}f(x).$

Next, we prove the $L^{2}-$estimate of the third term. By the covering
argument, we have that

$\displaystyle\left\|\sup_{k\in{\mathbb{Z}}}\sup_{v\in
T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{2}$
$\displaystyle\lesssim$
$\displaystyle\left(\sum_{k\in{\mathbb{Z}}}\left\|\sup_{v\in
T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{2}^{2}\right)^{\frac{1}{2}}$
$\displaystyle\lesssim$
$\displaystyle\left(\sum_{k\in{\mathbb{Z}}}2^{(j+l)s}\left\|\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{2}^{2}\right)^{\frac{1}{2}}$
(8.10) $\displaystyle\lesssim$ $\displaystyle
2^{\frac{ls}{2}}2^{-j\frac{d-2-s}{2}}\|f\|_{2}.$

Interpolating the $L^{1}$ from (8.9) and $L^{2}-$estimate (8.10), we get that
for $1<p\leq 2$

$\left\|\sup_{k\in{\mathbb{Z}}}\sup_{v\in T_{2}}\sup_{2^{-l}<r\leq
2^{-l+1}}|f\ast\psi_{2^{k-j}r}\ast\sigma_{2^{k}r}(\cdot+2^{k}v)|\right\|_{p}\lesssim
2^{-j(d-s-1-\frac{d-s}{p})}2^{l(\frac{2d-s}{p}-d+s)}\|f\|_{p}.$

Then, summing in $j$ concludes the proof of Lemma 8.2. ∎

We now complete the proof of the boundedness of $\mathcal{N}^{T}_{lac}$ at
points on the line segment $AQ$, excluding the point $Q$. By Lemma 8.2 and the
weak type $(1,1)$ boundedness of $M_{l}$, we get that

$\|\mathcal{N}_{lac}^{T}(f,g)\|_{\frac{p}{p+1},\infty}\lesssim\sum_{l=1}^{\infty}2^{-l(d-2)}2^{l(\frac{2d-s}{p}-d+s)}\|f\|_{1}\|g\|_{p}\lesssim\|f\|_{1}\|g\|_{p},$

where the series in $l$ is summable for $p>1+\frac{1}{d-s-1}.$ This completes
the proof of Theorem 4.1.∎

##  9\. Proof of Theorem 4.2: Lacunary bilinear maximal operator
$\mathcal{N}_{lac}^{T}$ in dimension $d=1$

We consider a new (sub)linear operator

$\mathbb{A}^{T}_{lac}f:=\sup_{u\in
T,k\in\mathbb{Z}}\Big{|}\int_{\mathbb{S}^{1}}f(x+2^{k}(u+y))~{}d\sigma(y,z)\Big{|}.$

######  Lemma 9.1.

Let $T$ be a compact subset of $\mathbb{R}$. Then $\mathbb{A}^{T}_{lac}$ is
bounded in $L^{p}({\mathbb{R}})$ for $1<p\leq\infty$, and
$\mathbb{A}^{T}_{lac}$ maps $L^{1}({\mathbb{R}})$ to
$L^{1,\infty}({\mathbb{R}})$.

###### Proof.

Note that we can rewrite the operator as above in the following way.

$\mathbb{A}^{T}_{lac}f(x)\simeq\sup_{u\in
T,k\in\mathbb{Z}}\int^{1}_{0}f(x+2^{k}(u+t))~{}\frac{dt}{\sqrt{1-t^{2}}}.$

Moreover, we can write

$\displaystyle\int^{1}_{0}f(x+2^{k}(u+t))~{}\frac{dt}{\sqrt{1-t^{2}}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{j=1}^{\infty}2^{\frac{j}{2}}\int_{1-2^{-(j-1)}}^{1-2^{-j}}f(x+2^{k}(u+t))~{}dt$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\frac{2^{-\frac{j}{2}}}{|I_{j}|}\int_{I_{j}}f(x+2^{k}(u+t))~{}dt,$

where $I_{j}=[1-2^{-(j-1)},1-2^{-j}].$ From estimates of shifted dyadic Hardy-
Littlewood maximal function[Muscalu], we get weak type (1,1) estimates for
$\mathbb{A}^{T}_{lac}$. Now interpolating with the trivial
$L^{\infty}({\mathbb{R}})-$estimate, we get the desired lemma. ∎

###### Proof of Theorem 4.2.

abc
Proof of boundedness in the region $\Omega(\\{O,A,B\\})$: As in the proof of
Theorem 4.1, it suffices to prove weak-type estimates at the points $A$ and
$B$. Observe that

$\displaystyle\mathcal{N}^{T}_{lac}(f,g)(x)\leq\|f\|_{\infty}\mathbb{A}^{T}_{lac}g(x),~{}~{}\mathcal{N}^{T}_{lac}(f,g)(x)\leq\|g\|_{\infty}\mathbb{A}^{T}_{lac}f(x).$

Therefore, using Lemma 9.1 we get that for $0\leq s\leq 1$,
$\mathcal{N}^{T}_{lac}$ maps $L^{p_{1}}(\mathbb{R})\times
L^{p_{2}}(\mathbb{R})$ to $L^{p_{3},\infty}(\mathbb{R})$ for
$(\frac{1}{p_{1}},\frac{1}{p_{2}})\in\\{A,B\\}$.
Proof of boundedness in the region $\Omega(\\{O,A,D,B\\})$: It remains to
prove weak type estimates for points on the diagonal $OD,$ excluding the point
$D$ for $0\leq s<\frac{1}{2}$. Using the identity (5.1) we get for any given
$k\in\mathbb{Z}$,

$\mathcal{N}^{T}_{2^{k}}(f,g)(x)\leq\mathcal{A}^{\phi,\phi}_{2^{k}}(f,g)(x)+\mathcal{A}^{\phi,\infty}_{2^{k}}(f,g)(x)+\mathcal{A}^{\infty,\phi}_{2^{k}}(f,g)(x)+\sum_{i,j\geq
1}\mathcal{A}^{i,j}_{2^{k}}(f,g)(x).$

Observe that

$\displaystyle\sup_{k\in\mathbb{Z}}\mathcal{A}^{\phi,\phi}_{2^{k}}(f,g)(x)\lesssim
M_{HL}f(x)M_{HL}(g)(x),~{}~{}\sup_{k\in\mathbb{Z}}\mathcal{A}^{\phi,\infty}_{2^{k}}(f,g)(x)\lesssim
M_{HL}f(x)\textbf{A}^{T}_{lac}g(x)$ and
$\displaystyle\sup_{k\in\mathbb{Z}}\mathcal{A}^{\infty,\phi}_{2^{k}}(f,g)(x)\lesssim\textbf{A}^{T}_{lac}f(x)M_{HL}g(x).$

Therefore, we only need to prove desired estimates for
$\mathcal{A}^{i,j}_{2^{k}}$. The arguments used to deal with the higher
dimensional case can be used to prove that

(9.1) $\displaystyle\|\mathcal{A}^{i,j}_{1}(f,g)\|_{L^{1/2}}\leq
C2^{(i+j)s}\|f\|_{L^{1}}\|g\|_{L^{1}},$

where

$\mathcal{A}^{i,j}_{1}(f,g)(x)=\sup_{u,v\in
T}\Big{|}\int_{\mathbb{S}^{1}}f\ast\psi_{2^{-i}}(x+u+y)g\ast\psi_{2^{-j}}(x+v+z)~{}d\sigma(y,z)\Big{|}.$

Note that
$|\widehat{\sigma_{\mathbb{S}^{1}}}(\xi,\eta)|\lesssim(1+|(\xi,\eta)|)^{-1/2}$.
Therefore, from Lemma 8.1 we get that

(9.2) $\displaystyle\|\mathcal{A}^{i,j}_{1}(f,g)\|_{L^{1}}\lesssim
2^{-(i+j)/4}2^{(i+j)\frac{s}{2}}\|f\|_{L^{2}}\|g\|_{L^{2}}.$

Now summing over $i,j$ we get $L^{2}(\mathbb{R})\times L^{2}(\mathbb{R})$ to
$L^{1}(\mathbb{R})$ boundedness when $s<\frac{1}{2}$. Further, interpolating
the estimates (9.1) and (9.2) we get

$\|\mathcal{A}^{i,j}_{1}(f,g)\|_{L^{p/2}}\lesssim
2^{\frac{(i+j)s}{p}}2^{-\frac{(i+j)}{2}(1-\frac{1}{p})}\|f\|_{L^{p}}\|g\|_{L^{p}}.$

Finally, applying the vector-valued argument as in Section 8 we get
$L^{p}(\mathbb{R})\times L^{p}(\mathbb{R})\rightarrow
L^{\frac{p}{2}}(\mathbb{R})$ boundedness of $\mathcal{N}^{T}_{lac}$ for
$\frac{1}{p}<\frac{1}{1+2s}$, when $s<1/2$. ∎

##  10\. Necessary conditions

In this section, we construct an example to show the sharpness of the
restricted weak type inequality $A_{1}^{T}:L^{1+\frac{s}{d-1},r}\to
L^{1+\frac{s}{d-1},\infty}$. The example is a modification of the arguments
given in [EndpointMappingPropertiesOfSphericalMaximalOperators], where they
showed the sharpness in the particular case when $T$ is of Minkowski dimension
$\frac{1}{2}$ in dimension $d=2$.

To obtain the counterexample, we will need two self-similar sets of
appropriate Minkowski content whose sum is an interval. We say $C$ is a Cantor
set of ratio $\lambda$ if $C$ is self-similar with respect to the similitudes
$S_{1}(x)=\lambda x$ and $S_{2}(x)=\lambda x-(1-\lambda)$ i.e.,
$C=S_{1}(C)\cup S_{2}(C)$. It is well-known that the Minkowski dimension of
$C$ is $\frac{\log 2}{\log\frac{1}{\lambda}}$, see Section 2.2 of Chapter 7 in
[SSbook]. The following lemma justifies the existence of two Cantor sets whose
sum is an interval.

######  Lemma 10.1.

Let $C_{1}$ and $C_{2}$ be two Cantor sets with ratios $\lambda_{1}$ and
$\lambda_{2}$ respectively such that
$\frac{\lambda_{1}}{1-2\lambda_{1}}\frac{\lambda_{2}}{1-2\lambda_{2}}\geq 1$.
Then $C_{1}+C_{2}$ is a closed interval. In fact, we have $C_{1}+C_{2}=[0,2]$.

We refer the interested reader to [Theorem 1.1, [Takahashi]] for more details.

######  Proposition 10.2.

For $r>1$, the operator $A_{1}^{T}$ does not map $L^{1+\frac{s}{d-1},r}$ to
$L^{1+\frac{s}{d-1},\infty}$.

###### Proof.

Let $C_{1},C_{2}\subset[0,1]$ be Cantor sets of ratios $2^{-\frac{1}{s}}$ and
$2^{-\frac{1}{1-s}}$ respectively. Then from Lemma 10.1 we can see that
$[-1,1]=C_{2}-C_{1}=\\{c_{2}-c_{1}:c_{1}\in C_{1},c_{2}\in C_{2}\\}$. For a
small $a>0$, define

$f(x)=\sum_{i=1}^{N}4^{(d-1)i}\chi_{\\{ce_{1}:c\in C_{2}\\}+B(0,a4^{-i})}(x).$

For $r>1$, we have the following bound on Lorentz space norm of $f$,

$\|f\|_{L^{1+\frac{s}{d-1},r}}\lesssim N^{\frac{1}{r}}.$

To see this, consider the sets
$F_{j}=\bigg{(}4^{(d-1)j}\sum\limits_{k=0}^{j-1}4^{-(d-1)k},4^{(d-1)(j+1)}\sum\limits_{k=0}^{j}4^{-(d-1)k}\bigg{]}$
for $1\leq j\leq N-1$. If $s\in F_{j}$ , we have

$\\{x\in{\mathbb{R}}^{d}:\;|f(x)|>s\\}=\bigcup\limits_{y\in
C}y+B(0,a4^{-(j+1)}).$

Denote $d_{f}(s)=s|\\{x\in{\mathbb{R}}^{d}:\;|f(x)|>s\\}|$ and observe that,

$sd_{f}(s)^{\frac{1}{p}}=s[(a4^{-(j+1)})^{d-(1-s)}]^{\frac{d-1}{d-1+s}}\lesssim
1.$

Therefore, we have

$\displaystyle\|f\|_{L^{1+\frac{s}{d-1},r}}$
$\displaystyle=\left(\frac{d-1+s}{d-1}\right)^{\frac{1}{q}}\left(\int_{0}^{4^{d-1}}[d_{f}(s)s]^{q}\frac{ds}{s}+\sum_{j=0}^{N-1}\int_{F_{j}}[d_{f}(s)s]^{q}\frac{ds}{s}\right)^{\frac{1}{r}}$
$\displaystyle\lesssim\left(\int_{0}^{4^{d-1}}4^{-(d-1)}s^{q-1}\;ds+\sum_{j=0}^{N-1}\int_{F_{j}}\frac{ds}{s}\right)^{\frac{1}{r}}$
$\displaystyle\leq\left(4^{(q-1)(d-1)}+\sum_{j=0}^{N-1}\right)^{\frac{1}{q}}\lesssim
N^{\frac{1}{r}}$

Let $T=\\{ce_{1}:\;c\in C_{1}\\}$. Now for $x\in[-1,1]e_{1}+\mathbb{S}^{d-1}$,
there exists $u\in T$ and $y\in\mathbb{S}^{d-1}$ such that

$x+u+y=z\in\\{ce_{1}:c\in C_{2}\\}\text{\;\;and\;\;}|\mathbb{S}^{d-1}(x+u)\cap
B(z,a4^{-i})|\simeq 4^{-i(d-1)}.$

Thus, we can see that $A^{T}_{1}f(x)\gtrsim N$. Now, suppose $A^{T}_{1}$ maps
$L^{1+\frac{s}{d-1},r}$ to $L^{1+\frac{s}{d-1}}$, then

$\displaystyle 1\lesssim|\\{x:A^{T}_{1}f(x)\gtrsim
N\\}|\lesssim\left(\frac{\|A^{T}_{1}\|_{L^{1+\frac{s}{d-1},r}\to
L^{1+\frac{s}{d-1},\infty}}\|f\|_{L^{1+\frac{s}{d-1},r}}}{N}\right)^{1+\frac{s}{d-1}}\lesssim\frac{\|A^{T}_{1}\|_{L^{1+\frac{s}{d-1},r}\to
L^{1+\frac{s}{d-1},\infty}}}{N^{\left(1-\frac{1}{r}\right)\left(1+\frac{s}{d-1}\right)}}.$

The desired result follows by taking the limit $N\to\infty$. ∎

## Acknowledgement

Ankit Bhojak and Saurabh Shrivastava acknowledge the financial support from
Science and Engineering Research Board, Department of Science and Technology,
Govt. of India, under the scheme Core Research Grant, file no.
CRG/2021/000230. Surjeet Singh Choudhary is supported by CSIR(NET), file
no.09/1020(0182)/2019- EMR-I for his Ph.D. fellowship. Kalachand Shuin is
supported by NRF grant no. 2022R1A4A1018904 and BK 21 Post doctoral
fellowship. The authors acknowledge the support and hospitality provided by
the International Centre for Theoretical Sciences, Bangalore (ICTS) for
participating in the program - Modern trends in Harmonic Analysis (code:
ICTS/Mtha2023/06).

## References