Hilbert boundary value problems—a distributional approach

Abstract

Synopsis Hilbert boundary value problems for a half-space are considered for analytic representations of Schwartz distributions: given data g ∈ D '(ℛ) and a coefficient x we seek functions F ( z ) analytic for Jmz ≠0 whose limits exist in D '(ℛ) and satisfy F + — X F – = g on an open subset U of the real line R. U is the complement of a finite set which contains the singular support and the zeros of X · X and its reciprocal satisfy certain growth conditions near the boundary points of U . Solutions F ( z ) are shown to exist, and their general form is determined by obtaining a suitable factorisation of x .