Boundary estimates for the fractional spherical maximal function

TL;DR

This paper investigates the boundary conditions for the boundedness of fractional spherical maximal functions and establishes weak type estimates at the critical boundary regions.

Contribution

It provides necessary and sufficient conditions for $L^p-L^q$ boundedness and proves boundary weak type estimates for both full and lacunary fractional spherical maximal functions.

Findings

Established boundary weak type estimates for fractional spherical maximal functions.

Derived necessary and sufficient conditions for $L^p-L^q$ boundedness.

Analyzed both full and lacunary versions of the maximal functions.

Abstract

In this article, we study the fractional spherical maximal function and its lacunary counterpart. We study the necessary and sufficient conditions for $L^p-L^q$ boundedness of both maximal functions. In particular, we prove the restricted weak type estimate for both full and lacunary fractional spherical maximal functions at the boundary of the maximal $L^p-L^q$ bounded regions.