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 -Delta u = f, where is an element of M ((H) over dot(1)(R(d)) -> (H) over dot(-1)(R(d))) is a complex-valued distribution on R(d), satisfy the regularity property D(k) u is an element of M ((H)over dot(1) -> (H)over dot(-1)) for all k, vertical bar k vertical bar = 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.