On Isomorphism of Some Functional Spaces under Action of Two-dimensional Singular Integral Operators

Abstract

AbstractWe address two-dimensional singular integral equations widely used for constructing and investigating the solutions to the general linear first-order elliptic systems in the 2D domains. Proving the solvability of such integral equations with the use of the Calderón–Zygmund theorem brings us at the statements about the existence of the solution belonging to the spaces of summable functions whose summability exponents have to be close to the value of two, and the increase of regularity of the problem’s data does not eliminate this restriction automatically. In this article, we prove a regularity result for the solutions to two-dimensional singular integral equations with the use of the representations of the second kind for the solutions to the first-order general linear elliptic systems discovered by the author in his prior work.KeywordsTwo-dimensional singular integral equationsMathematics Subject Classification45E99 + 30E20 + 44A15