On Rubio de Francia's maximal theorem

TL;DR

This paper extends Rubio de Francia's 1986 results on maximal functions generated by measures with Fourier decay, providing endpoint bounds, local estimates, and analyzing the impact of Frostman's growth condition.

Contribution

It offers new endpoint and local bounds for the maximal function and explores the influence of Frostman's growth condition beyond Fourier decay.

Findings

Established restricted weak-type endpoint bounds.

Derived $L^p$--$L^q$ bounds for local maximal variants.

Showed Frostman's condition affects bounds when decay exceeds $d-1$.

Abstract

In his influential 1986 paper, Rubio de Francia established $L^p$ bounds for the maximal function generated by dilations of measures $\mu$ whose Fourier transforms $\widehat{\mu}$ 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 $\widehat{\mu}$, 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.