Analysis in the multi-dimensional ball

TL;DR

This paper investigates the heat semigroup maximal operator in a d-dimensional ball, establishing Gaussian bounds for the heat kernel and deriving weighted $L^p$ and mixed norm inequalities for the operator.

Contribution

It provides new Gaussian bounds for the heat kernel and proves weighted inequalities for the associated maximal operator in a multi-dimensional setting.

Findings

Gaussian bounds for the heat kernel in the d-dimensional ball

Weighted $L^p$ estimates for the maximal operator

Weighted inequalities in mixed norm spaces

Abstract

We study the heat semigroup maximal operator associated with a well-known orthonormal system in the d-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted $L^p$ estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.