Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bilinear spherical maximal function with fractal dilations

Surjeet Singh Choudhary, Chun-Yen Shen, Saurabh Shrivastava

TL;DR

This work quantifies the $L^p$ boundedness of the bilinear spherical maximal function with fractal dilations $\mathcal{M}_E$ in terms of the dilation-set’s upper Minkowski dimension $\beta$, establishing sharp-type ranges that depend on the ambient dimension $d$ and fractal size. The authors develop a scale-adaptive, multiplier-centric framework that combines a refined Littlewood-Paley decomposition, a detailed analysis of the spherical surface measure via Bessel asymptotics, and bilinear Calderón-Zygmund theory to control both local and global maximal operators. They obtain novel bounds for the lacunary case $\beta=0$ in $dimensions\;d\ge4$ that extend known $L^p$ ranges to borderline exponents, including end-point restricted weak-type estimates along specific line segments. The results unify and extend previous work on $M_E$, the local versus global dichotomy, and the lacunary bilinear maximal function, offering a fractal-geometric perspective on multilinear averaging over spheres with general dilations.

Abstract

In this paper, we investigate $L^p-$boundedness of the bilinear spherical maximal function associated with a general dilation set $E\subset\R_+$. We quantify the range of $L^p-$boundedness in terms of a dilation-invariant notion of upper Minkowski dimension of the set $E$. A particular case of this study settles an open question of $L^p-$boundedness of the lacunary bilinear spherical maximal function at borderline cases $p_1=1$ or $p_2=1$ in dimension $d\geq4$.

Bilinear spherical maximal function with fractal dilations

TL;DR

This work quantifies the boundedness of the bilinear spherical maximal function with fractal dilations in terms of the dilation-set’s upper Minkowski dimension , establishing sharp-type ranges that depend on the ambient dimension and fractal size. The authors develop a scale-adaptive, multiplier-centric framework that combines a refined Littlewood-Paley decomposition, a detailed analysis of the spherical surface measure via Bessel asymptotics, and bilinear Calderón-Zygmund theory to control both local and global maximal operators. They obtain novel bounds for the lacunary case in that extend known ranges to borderline exponents, including end-point restricted weak-type estimates along specific line segments. The results unify and extend previous work on , the local versus global dichotomy, and the lacunary bilinear maximal function, offering a fractal-geometric perspective on multilinear averaging over spheres with general dilations.

Abstract

In this paper, we investigate boundedness of the bilinear spherical maximal function associated with a general dilation set . We quantify the range of boundedness in terms of a dilation-invariant notion of upper Minkowski dimension of the set . A particular case of this study settles an open question of boundedness of the lacunary bilinear spherical maximal function at borderline cases or in dimension .

Paper Structure

This paper contains 12 sections, 14 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bilinear spherical maximal function
  3. Main results
  4. Strategy of proof
  5. Reduction using Littlewood-Paley decomposition
  6. Decomposition of multiplier
  7. Estimate for local maximal operator
  8. Proof of \ref{['l2']}.
  9. Proof of \ref{['l1']}.
  10. Estimate for global maximal operator: Proof of \ref{['main']}
  11. Proof of \ref{['L2']}.
  12. Proof of parts $(1)$ and $(2)$ of \ref{['L1']}

Key Result

Theorem A

ChristZhouBorgesFoster Let $1\leq p_1,p_2\leq\infty$ with $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. Then, $\mathcal{M}_{lac}$ maps $L^{p_1}(\mathbb{R}^{d})\times L^{p_2}(\mathbb{R}^{d})$ to $L^{p}(\mathbb{R}^{d})$ if Moreover, the weak-type bounds $\mathcal{M}_{lac}:L^{1}(\mathbb{R}^{d})\times L^{\infty}(\mathbb{R}^{d})\rightarrow L^{1,\infty}(\mathbb{R}^{d})$ and $\mathcal{M}_{lac}:L^{\infty}(\

Figures (3)

  • Figure 1: Boundedness regions for the operator $\mathcal{M}_{lac}$ in \ref{['lac1']}.
  • Figure 2: The dark gray region depicts the improved range for boundedness of $\mathcal{M}_E$.
  • Figure 3: Boundedness region for the operator $\mathcal{M}_{lac}$ in dimension $d\geq4$.

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem A
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2: HHY, Lemma 2.2
  • Lemma 5.1: Lee1, Lemma 2.6
  • ...and 6 more