On the Boundedness of Cauchy Singular Operator from the Space Lp to Lq , p > q ≥ 1

Abstract

Abstract It is proved that for a Cauchy type singular operator, given by equality (Khvedelidze, Current problems of mathematics 7: 5-162, 1975), to be bounded from the Lebesgue space Lp (Γ) to Lq (Γ), as , it is necessary and sufficient that either condition (Calderon, Proc. Nat. Acad. Sci. USA 4: 1324-1327, 1977) or (David, L'integrale de Cauchy sur le courbes rectifiables, Prepublications Univ. Paris–Sud; Dept. Math., 1982) be fulfilled.