An atomic representation for Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations

TL;DR

This paper introduces an atomic representation for Hardy class solutions to nonhomogeneous Cauchy-Riemann equations, enabling boundary value analysis, Hilbert transform continuity, and solution construction for related boundary problems.

Contribution

It develops a second-kind representation for Hardy class solutions, linking atomic decomposition with boundary value and Hilbert transform analysis.

Findings

Boundary values can be decomposed into atomic and error parts.

Hilbert transform is continuous on these distribution classes.

Solutions to Schwarz-type boundary problems can be constructed within Hardy classes.

Abstract

We develop a representation of the second kind for certain Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations and use it to show that boundary values in the sense of distributions of these functions can be represented as the sum of an atomic decomposition and an error term. We use the representation to show continuity of the Hilbert transform on this class of distributions and use it to show that solutions to a Schwarz-type boundary value problem can be constructed in the associated Hardy classes.