On Smirnov classes of harmonic functions, and the Dirichlet problem (Q2709543)

Two types of the Smirnov classes of harmonic functions are studied. Namely, \(l^{p}(D)\) is the class of those harmonic functions \(u\) such that the integral mean of \(|u|^{p}\) along the family of representing curves is uniformly bounded, and \({\widetilde l}^{p}(D) = \Re E^{p}(D)\), where \(E^{p}(D)\) are the Smirnov classes of analytic functions. Sufficient conditions for the equality NEWLINE\[NEWLINE l^{p}(D) = {\widetilde l}^{p}(D)NEWLINE\]NEWLINE are given. Examples of domains for which (1) is not valid are found. The Dirichlet problem in \(l^{p}(D), p > 1,\) are investigated in the case of domains having conformal representation of certain type. The conditions of solvability, unique solvability and nonsolvability of the above problem are presented in the form of the properties of conformal representation.