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 parallel to L alpha F parallel to(Ld/(d-alpha),1(Rd;Rd)) <= C parallel to F parallel to(L1(Rd;Rd)) for all fields F is an element of L-1 (R-d;R-d) such that 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. (C) 2020 The Author. Published by Elsevier Inc.