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On Rubio de Francia's maximal theorem

Seheon Ham, Jiwon Kah, Sanghyuk Lee, Ji Li

TL;DR

This work extends Rubio de Francia's maximal theorem by establishing a restricted weak-type endpoint at $p=p_a$ for maximal operators generated by dilations of fractal measures with Fourier decay, and by clarifying how Frostman growth (dimension) affects $L^p$ bounds when the Frostman exponent $b$ exceeds $d-1$. It introduces a refined threshold $p_{(a,b)}$ that governs $L^p$ boundedness in the $b>d-1$ regime and shows the Frostman condition is immaterial for $b\le d-1$, while yielding an endpoint bound there as well. The paper also analyzes the local maximal operator $M_μ^{loc}$, proving $L^p$-improving estimates from $L^{p_{(a,b)},1}$ to $L^{p'_{(a,b)},\infty}$ and demonstrating further range enlargement via dispersive/Strichartz-type estimates. Methodologically, it combines a dyadic multiplier decomposition, Hardy space and $H^1$–$L^1$ control, Littlewood–Paley theory, Bourgain's summation trick, and $TT^*$-type dispersive arguments to achieve sharp endpoint results and sharp range descriptions. Collectively, the results deepen the understanding of maximal averaging over fractal measures and illuminate how geometric growth and decay conditions interact with $L^p$–$L^q$ mapping properties in harmonic analysis.

Abstract

In his influential 1986 paper, Rubio de Francia established $L^p$ bounds for the maximal function generated by dilations of measures $μ$ whose Fourier transforms $\widehatμ$ satisfy specific decay condition. In the present work, we obtain results that complement his work in several directions. In particular, we obtain restricted weak-type endpoint bound on the maximal function and $L^p$--$L^q$ bounds on its local variant. We also investigate how Frostman's growth condition on the measure influences those maximal bounds. While a key feature of Rubio de Francia's result is that $L^p$ boundedness is determined solely by the decay order of $\widehatμ$, we show that the Frostman condition plays a significant role when the growth order exceeds $d-1$ or when $L^p$--$L^q$ estimates are considered.

On Rubio de Francia's maximal theorem

TL;DR

This work extends Rubio de Francia's maximal theorem by establishing a restricted weak-type endpoint at for maximal operators generated by dilations of fractal measures with Fourier decay, and by clarifying how Frostman growth (dimension) affects bounds when the Frostman exponent exceeds . It introduces a refined threshold that governs boundedness in the regime and shows the Frostman condition is immaterial for , while yielding an endpoint bound there as well. The paper also analyzes the local maximal operator , proving -improving estimates from to and demonstrating further range enlargement via dispersive/Strichartz-type estimates. Methodologically, it combines a dyadic multiplier decomposition, Hardy space and – control, Littlewood–Paley theory, Bourgain's summation trick, and -type dispersive arguments to achieve sharp endpoint results and sharp range descriptions. Collectively, the results deepen the understanding of maximal averaging over fractal measures and illuminate how geometric growth and decay conditions interact with – mapping properties in harmonic analysis.

Abstract

In his influential 1986 paper, Rubio de Francia established bounds for the maximal function generated by dilations of measures whose Fourier transforms satisfy specific decay condition. In the present work, we obtain results that complement his work in several directions. In particular, we obtain restricted weak-type endpoint bound on the maximal function and -- bounds on its local variant. We also investigate how Frostman's growth condition on the measure influences those maximal bounds. While a key feature of Rubio de Francia's result is that boundedness is determined solely by the decay order of , we show that the Frostman condition plays a significant role when the growth order exceeds or when -- estimates are considered.
Paper Structure (20 sections, 16 theorems, 191 equations, 1 figure)

This paper contains 20 sections, 16 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Endpoint estimate for $p=p_a$
  3. Frostman condition
  4. Local maximal function
  5. Proof of Theorem \ref{['endp']}
  6. Decomposition of $\mu$ and $L^2$ estimate
  7. Estimate with a $H^1(\mathbb{R}^d)$ atom
  8. Preliminary decomposition
  9. Estimate for $I_{\mathrm{near}}$
  10. Estimate for $I_{\mathrm{far}}$
  11. Estimate for $I_{\mathrm{far}}^{(1)}$
  12. Estimate for $I_{\mathrm{far}}^{(2)}$
  13. Proof of Theorem \ref{['decay+dim']}
  14. Decomposition via Littlewood--Paley decomposition
  15. Proof of $L^2$ estimate \ref{['mjL2']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose that $\mu$ is a compactly supported Borel measure and with $a >\frac{1}{2}$. Then, for all $p >p_a := \frac{ 2a+1}{2a}$, we have

Figures (1)

  • Figure 1: The $L^p$-improving region $\Delta$ described in Theorem \ref{['pqr']}, formed by the convex hull of $O, P, Q, R$.

Theorems & Definitions (27)

  • Theorem 1.1: Ru86
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof : Proof of \ref{['eq:gt-majorized']}.
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 17 more