Hardy spaces of the conjugate Beltrami equation

TL;DR

This paper investigates Hardy spaces related to the conjugate Beltrami equation with Lipschitz coefficients on smooth planar domains, analyzing boundary behavior, trace density, and establishing an analog of Fatou's theorem for related boundary value problems.

Contribution

It introduces a detailed analysis of Hardy spaces for the conjugate Beltrami equation, including boundary behavior and a Fatou theorem analog for Dirichlet and Neumann problems.

Findings

Boundary behavior of Hardy space solutions characterized.

Density properties of traces established.

Fatou theorem analog derived for boundary value problems.

Abstract

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents $1<\infty$. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation ${div}(\sigma\nabla u)=0$ with $L^p$-boundary data.