Endpoint estimates and optimality for the generalized spherical maximal operator on radial functions

TL;DR

This paper establishes sharp boundedness conditions and endpoint estimates for a generalized spherical maximal operator acting on radial functions within weighted Lebesgue spaces, advancing understanding of its optimal behavior.

Contribution

It provides the first sharp conditions and endpoint results for the generalized spherical maximal operator on radial functions in weighted Lebesgue spaces.

Findings

Sharp boundedness conditions derived for the operator.

Optimal weak and restricted weak type endpoint estimates obtained.

Results specify precise weight and exponent ranges for boundedness.

Abstract

We find sharp conditions for the maximal operator associated with generalized spherical mean Radon transform on radial functions $M^{\a,\b}_t$ to be bounded on power weighted Lebesgue spaces. Moreover, we also obtain the corresponding endpoint results in terms of optimal power weighted weak and restricted weak type estimates.