An Optimal Sobolev Embedding for $L^1$

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 \[ \|I_αF \|_{L^{d/(d-α),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C \|F\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \] for all fields $F\in L^1({\mathbb{R}}^d;{\mathbb{R}}^d)$ such that $\mathrm{curl}F=0$ 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 $p>1$.

19 pages