$L^p$ estimates for angular maximal functions associated with Stieltjes and Laplace transforms

TL;DR

This paper establishes $L^p$ estimates for maximal angular functions linked to Stieltjes and Laplace transforms, demonstrating their boundedness as nonlinear operators between $L^p$ and $L^q$ spaces.

Contribution

It introduces bounds for maximal angular operators composed with classical transforms, extending known $L^p$ estimates to these nonlinear maximal functions.

Findings

Boundedness of maximal angular operators with Poisson, Stieltjes, and Laplace transforms.

Extension of $L^p$ estimates to nonlinear maximal functions.

Operators are bounded from $L^p$ to $L^q$ for the same $p,q$ as classical transforms.

Abstract

Maximal angular operator sends a function defined in a sector of the complex plane to a Maximal angular operator sends a function defined in a sector of the complex plane with vertex at 0 to the function of modulus obtained by maximizing over argument. Compositions of the so defined maximal angular operator (in suitable sectors) with the Poisson, Stieltjes and Laplace transforms are shown to be bounded (nonlinear) operators from $L^p$ to $L^q$ for the same values of $p$ and $q$ as their standard counterparts.