On a family of integral operators on the ball

TL;DR

This paper studies a class of integral operators on the unit ball, deriving a Poisson kernel, establishing an extension inequality, and analyzing extremal functions with conformal invariance considerations.

Contribution

It transforms a known equation from the upper half space to the unit ball, identifies the Poisson kernel, and proves a new extension inequality with extremal function characterization.

Findings

Derived the Poisson kernel for the equation in the unit ball

Proved an extension inequality in the limit case

Established the uniqueness of extremal functions

Abstract

In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $\mathbb{B}^n$. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and prove the uniqueness of the extremal functions in the limit case using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.