Endpoint estimates for the fractal circular maximal function and related local smoothing

2025/06/25 by Sanghyuk Lee, Luz Roncal, Lee, Sanghyuk +4 · 3 citations
Mathematics · #Mathematical Dynamics and Fractals #Mathematical Approximation and Integration #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2506.20390

Abstract

Sharp L^p--L^q estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson--Hughes--Roos--Seeger. More intricate L^p--L^q estimates for the fractal circular maximal function were later established in the sharp range by Roos--Seeger, but the endpoint estimates have been left open, particularly when the fractal dimension of the dilation set lies in [1/2, 1). In this work, we prove these missing endpoint estimates for the circular maximal function. We also study the closely related L^p--L^q local smoothing estimates for the wave operator over fractal dilation sets, which were recently investigated by Beltran--Roos--Rutar--Seeger and Wheeler. Making use of a bilinear approach, we also extend the range of p,q, for which the optimal estimate holds.

Cited by

• Problems on spherical maximal functions
• Bilinear spherical maximal function with fractal dilations
• Strichartz and local smoothing estimates for the fractional Schrödinger equations over fractal time

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• Bilinear spherical maximal function with fractal dilations
• Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets
• Sharp L² estimate of Schrödinger maximal function in higher dimensions