Adams inequalities on measure spaces

TL;DR

This paper generalizes Adams inequalities from bounded domains in R^n to arbitrary finite measure spaces, introducing new integral operators and applications in elliptic operators and CR geometry.

Contribution

It extends Adams' sharp inequalities to measure spaces with general integral operators, broadening their applicability and providing new sharp inequalities and applications.

Findings

Extended Adams inequalities to measure spaces with finite measure.

Introduced general integral operators satisfying explicit growth conditions.

Provided new applications including trace inequalities and CR setting operators.

Abstract

In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R^n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.