Hardy--Littlewood--Sobolev inequality for $p=1$

TL;DR

This paper establishes a Hardy--Littlewood--Sobolev type inequality for Riesz potentials acting on certain invariant distribution spaces, extending classical results to a broader functional framework with applications to elliptic operators.

Contribution

It proves a new inequality for Riesz potentials on invariant distribution subspaces that do not contain delta distributions, generalizing classical Sobolev inequalities.

Findings

The inequality holds for functions in specific invariant subspaces without delta distributions.

The result applies to derivatives of functions controlled by canceling elliptic operators.

A new Lorentz space estimate for Riesz potentials is established.

Abstract

Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type $a\otimes \delta_0$, $\delta_0$ being the Dirac delta, then the inequality $\|\mathbb{I}_\alpha [f]\|_{L_{p,1}}\lesssim \|f\|_{L_1}$, $\frac{p-1}{p} = \frac{\alpha}{d}$, holds true for functions $f\in\mathcal{W}\cap L_1$ with a uniform constant; here $\mathbb{I}_\alpha$ is the Riesz potential of order $\alpha$ and $L_{p,1}$ is the Lorentz space. This result implies as a particular case the inequality $\|\nabla^{m-1} f\|_{L_{\frac{d}{d-1},1}} \lesssim \|A f\|_{L_1}$, where $A$ is a canceling elliptic differential operator of order $m$.