Two-Dimensional Homogenous Integral Operators and Singular Operators with Measurable Coefficients in Fibers

Abstract

We study a new class of homogeneous operators in L₂(\( {\mathbb{R}}^2 \)) that, after foliation of \( {\mathbb{R}}^2 \) into concentric circles, are represented in fibres as singular integral operators with measurable essentially bounded coefficients. We find necessary and sufficient conditions for the invertibility of such operators and construct the operator-valued symbolic calculus for the C^∗–algebra generated by such operators and operators of multiplication by multiplicatively weakly oscillating functions. We obtain a criterion for the generalized Fredholm property of operators and find effectively verifiable functional necessary conditions for the classical Fredholm property.