About decidability of singular integral equation in Besov space

Abstract

The existence and uniqueness of solutions of singular integral equation in Lp(E), p > 2 obtained by V.S. Vinogradov in the paper "On the solvability of a singular integral equation". These results continued by I.I. Komyak in Lp(E), p > 1 in the case of a more general equation in the paper "On the solvability of a class of two-dimensional singular integral equations". And in this article studied the solubility of a singular integral equation in Besov spaces Bα p,1 (E), 1 < p < 2, α = 2 p − 1 but not embedded in Lq(E) not any q > 2. Solvability of this equation is equivalent to continuous solvability of the differential Beltrami equation ∂w ∂z −µ(z) ∂w ∂z = 0. Shown to be Noetherian solubility of singular integral equation, proved that the index is zero and the kernel consists only of zero.We explicitly construct the operator - regulyarizators have considered a singular integral. These results suggest the existence of a continuous homeomorphism of the Bel′ trami equation.