Hardy-Littlewood-Sobolev Theorem for Bourgain-Morrey Spaces and Approximation

Abstract

In this paper, we establish an extension of the Hardy-Littlewood-Sobolev theorem to the setting of the Bourgain-Morrey space M^α_q,_p(R^d) (1 ≤ q, p, α ≤ ∞), which theory goes back to Bourgain in 1991. We also prove that Mα q,p(R d ) is included in the closure of the Lebesgue space L^α in the Morrey-type space F(q, p, α), which arises naturally in 2015 in the study of boundedness properties of fractional integral operators. Therefore, we establish in M^α_q,p some approximation results by compactly supported and/or regular functions. As an application of these results, we obtain an explicit solution in [L^p(R^d)]^d of the equation div F = f whenever f is in M^α_q,p, with d ≥ 3, 1 ≤ q ≤ α < d and 1/p = 1/α – 1/d.