# Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on R^n

## TL;DR

This paper establishes sharp inequalities for Riesz potentials and fractional Laplacians on R^n, extending classical results to more general operators and measures, with implications for critical Sobolev spaces.

## Contribution

It derives sharp Adams and Moser-Trudinger inequalities for fractional and higher order operators on R^n, generalizing previous results to broader classes of potentials and measures.

## Key findings

- Sharp Adams inequalities for Riesz potentials on R^n.
- Sharp Moser-Trudinger inequalities for critical Sobolev spaces.
- Extension to fractional Laplacians and general elliptic operators.

## Abstract

We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of R^n. As a consequence, we obtain sharp Moser-Trudinger inequalities for the critical Sobolev spaces W^{a,n/a}(R^n), 0<a<n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.