A new family of sharp conformally invariant integral inequalities

TL;DR

This paper introduces a new family of sharp, conformally invariant integral inequalities on the unit ball, extending classical inequalities to higher dimensions and harmonic functions.

Contribution

It establishes a one-parameter family of sharp inequalities that generalize Carleman's inequality to poly-harmonic functions in higher dimensions.

Findings

Sharp inequalities are attained by conformal transformations.

The inequalities generalize classical results to higher dimensions.

A limiting case extends Carleman's inequality to poly-harmonic functions.

Abstract

We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant function under conformal transformations. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions.