Regularity of solutions of Poisson’s equation in multiplier spaces

Abstract

The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ $$ \mathcal{M} $$ (Ḣ1(ℝ d ) → (Ḣ−1(ℝ d )) is a complex-valued distribution on ℝ d , satisfy the regularity property D k u ∈ $$ \mathcal{M} $$ (Ḣ1 → Ḣ−1) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.