An Optimal Sobolev Embedding for $L^1$

TL;DR

This paper proves an optimal Lorentz space estimate for the Riesz potential on curl-free vector fields in , establishing the best possible bound in the critical case p=1, completing the known theory.

Contribution

It provides the first sharp Lorentz space estimate for Riesz potentials acting on curl-free fields at the critical exponent p=1.

Findings

Established the optimal Lorentz space bound for Riesz potentials on curl-free fields.

Completed the theory for the critical case p=1 in Sobolev embeddings.

Proved the estimate is sharp and cannot be improved.

Abstract

In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(\alpha,d)>0$ such that \[ \|I_\alpha F \|_{L^{d/(d-\alpha),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C \|F\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \] for all fields $F \in L^1(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{curl} F=0$ in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$.