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

TL;DR

This paper establishes the missing endpoint $L^p$--$L^q$ estimates for the fractal circular maximal function and extends local smoothing estimates for the wave operator over fractal dilation sets using a bilinear approach.

Contribution

It proves the previously open endpoint estimates for the fractal circular maximal function and broadens the range of $p,q$ for local smoothing estimates with a bilinear method.

Findings

Proved endpoint $L^p$--$L^q$ estimates for the fractal circular maximal function.

Extended the $p,q$ range for local smoothing estimates of the wave operator.

Utilized a bilinear approach to achieve these results.

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.