The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ \(
\)(Ḣ1(ℝd) → (Ḣ−1(ℝd)) is a complex-valued distribution on ℝd, satisfy the regularity property Dku ∈ \(
\)(Ḣ1 → Ḣ−1) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky  in the case where f belongs to the class of positive Borel measures.
Key words and phrases
Poisson equation regularity of solutions multiplier space
Mathematics subject classification numbers
35F05 42B15 42B35
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