# Adams inequalities for Riesz subcritical potentials

## TL;DR

This paper extends Adams inequalities to Riesz subcritical potentials on general measure spaces, introducing a new critical integrability condition and applying results to various geometric settings.

## Contribution

The authors develop new Adams inequalities for Riesz subcritical potentials with a novel integrability condition, broadening applicability to diverse measure spaces and domains.

## Key findings

- Derived sharp Adams inequalities on ${
m f R}^n$ and hyperbolic space.
- Established inequalities on Riesz subcritical domains with growth conditions.
- Extended results to domains satisfying Poincaré inequalities.

## Abstract

We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results obtained by the authors. The integral operators involved, which we call "Riesz subcritical", have kernels whose decreasing rearrangements are not worse than that of the Riesz kernel on ${\mathbb R}^n$, where the kernel is large, but they behave better where the kernel is small. The new element is a "critical integrability" condition on the kernel at infinity. Typical examples of such kernels are fundamental solutions of nonhomogeneous differential, or pseudo-differential, operators. Another example is the Riesz kernel itself restricted to suitable measurable sets, which we name "Riesz subcritical domains". Such domains are characterized in terms of their growth at infinity. As a consequence of the general results we obtain several new sharp Adams and Moser-Trudinger inequalities on ${\mathbb R}^n$, on the hyperbolic space, on Riesz subcritical domains, and on domains where the Poincar\'e inequality holds.