Critical Hardy inequality on the half-space via the harmonic transplantation

TL;DR

This paper establishes a critical Hardy inequality on the half-space using harmonic transplantation, improves the subcritical version, and discusses related Sobolev inequalities, advancing understanding of functional inequalities in half-space domains.

Contribution

Introduces a novel proof of the critical Hardy inequality on the half-space via harmonic transplantation and provides an improved subcritical inequality with convergence properties.

Findings

Proved a new critical Hardy inequality on the half-space.

Derived an improved subcritical Hardy inequality that converges to the critical case.

Discussed related Sobolev inequalities in the same setting.

Abstract

We prove a critical Hardy inequality on the half-space by using the harmonic transplantation. Also we give an improvement of the subcritical Hardy inequality on the half-space, which converges to the critical Hardy inequality. Sobolev type inequalities are also discussed.