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Maximal operators given by Fourier multipliers with dilation of fractional dimensions

Jin Bong Lee, Jinsol Seo

TL;DR

This work addresses obtaining $L^p$ bounds for maximal operators $\mathcal{M}_m^E$ formed from dilated Fourier multipliers $m(t\cdot)$ when the dilation set $E\subset(0,\infty)$ has fractional Minkowski dimension $\kappa(E)<1$. The authors develop a $\Sigma^2$-Besov regularity framework, proving $L^p$ boundedness for multipliers in $\Sigma^2(B_{p_0}^{d/p_0+s})$ with $\frac{1}{p_0}=\big|\frac{1}{2}-\frac{1}{p}\big|$ and $s>\frac{d}{p_0}+\frac{\kappa(E)}{2}$, via a vector-valued multiplier approach, fractional calculus, and bilinear interpolation. The results unify and extend existing criteria (including Mikhlin-type, limited decay, and slow decay multipliers) and provide a dimension-dependent mechanism linking the geometry of $E$ to the $p$-range of boundedness. This framework advances maximal Fourier multiplier theory in fractal-dilation settings and has potential applications to spherical-type averages and other fractal-dilation phenomena, where the dilation set's geometry critically governs $L^p$ behavior.

Abstract

In this paper, we investigate $L^p$ bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees $L^p$ bounds of the maximal operators for each $p$. Our criterion covers Mikhlin-type multipliers, multipliers with limited decay, and multipliers with slow decay.

Maximal operators given by Fourier multipliers with dilation of fractional dimensions

TL;DR

This work addresses obtaining bounds for maximal operators formed from dilated Fourier multipliers when the dilation set has fractional Minkowski dimension . The authors develop a -Besov regularity framework, proving boundedness for multipliers in with and , via a vector-valued multiplier approach, fractional calculus, and bilinear interpolation. The results unify and extend existing criteria (including Mikhlin-type, limited decay, and slow decay multipliers) and provide a dimension-dependent mechanism linking the geometry of to the -range of boundedness. This framework advances maximal Fourier multiplier theory in fractal-dilation settings and has potential applications to spherical-type averages and other fractal-dilation phenomena, where the dilation set's geometry critically governs behavior.

Abstract

In this paper, we investigate bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees bounds of the maximal operators for each . Our criterion covers Mikhlin-type multipliers, multipliers with limited decay, and multipliers with slow decay.
Paper Structure (10 sections, 8 theorems, 90 equations)

This paper contains 10 sections, 8 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of $\Sigma^2$ spaces
  4. Fractional calculus
  5. Proof of Theorem \ref{['thm_main']}
  6. Useful lemmas
  7. Proof of Theorem \ref{['thm_main']}
  8. Estimate \ref{['ineq_L2esti']}
  9. Estimate \ref{['ineq_endpts']}
  10. Proof of Lemma \ref{['lem_emb']}

Key Result

Theorem 1.1

Let $p\in(1,\infty)$ and $E\subset(0,\infty)$ satisfy $\kappa(E)<1$. Suppose that $m$ is of class $\Sigma^2(B_{p_0}^s)$ with $\frac{1}{p_0}=\left|\frac{1}{2}-\frac{1}{p}\right|$ and Then $\mathcal{M}_m^E$ is bounded on $L^p(\mathbb{R}^d)$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1
  • Lemma 2.1
  • proof
  • Lemma 2.2: Lemma 2.1, LeeSeo_2023
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 3 more