# Weighted Inequalities for the Fractional Laplacian and the Existence of Extremals

## TL;DR

This paper improves classical fractional Laplacian inequalities using Besov norms and establishes the existence of extremals in cases beyond previous theorems, advancing the understanding of weighted inequalities.

## Contribution

It introduces enhanced inequalities involving Besov norms and proves the existence of extremals in new parameter regimes not covered by prior results.

## Key findings

- Improved Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with Besov norms
- Existence of extremals in certain cases outside Lieb's theorem
- Advancement in weighted fractional Laplacian inequalities

## Abstract

In this article we obtain improved versions of Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein-Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of E. Lieb.