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

In this paper, we establish an extension of the Hardy-Littlewood-Sobolev theorem to the setting of the Bourgain-Morrey space Mαq,p(Rd) (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 Lα 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 Mαq,p some approximation results by compactly supported and/or regular functions. As an application of these results, we obtain an explicit solution in [Lp(Rd)]d of the equation div F = f whenever f is in Mαq,p, with d ≥ 3, 1 ≤ q ≤ α < d and 1/p = 1/α – 1/d.