Subcritical approach to conformally invariant extension operators on the upper half space

TL;DR

This paper establishes sharp embedding inequalities for conformally invariant extension operators on the upper half space, including classical Poisson and Riesz kernel operators, identifying extremal functions and their boundary behavior.

Contribution

It provides the first sharp constants and extremal functions for a broad family of conformally invariant extension operators, with detailed boundary analysis.

Findings

Sharp embedding inequalities obtained

Extremal functions classified and their boundary limits computed

Sharp constants are attained by specific extremal functions

Abstract

In this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators. This family includes among others the classical Poisson extension operator and the extension operator with Riesz kernel. We show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions. We also compute the limiting behavior at the boundary of the extensions of the extremal functions.