Regularity of solutions of Poisson’s equation in multiplier spaces

The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{M} $$\end{document}(Ḣ1(ℝd) → (Ḣ−1(ℝd)) is a complex-valued distribution on ℝd, satisfy the regularity property Dku ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{M} $$\end{document}(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.