Pseudo-holomorphic functions at the critical exponent

TL;DR

This paper investigates Hardy classes related to pseudo-holomorphic functions at the critical exponent, establishing new theorems and well-posedness results for boundary value problems with non-strictly elliptic coefficients.

Contribution

It introduces the first analysis of Hardy classes for the case r=2, proving an analog of the M. Riesz theorem and a topological converse to the Bers similarity principle.

Findings

Established an analog of the M. Riesz theorem for r=2.

Proved a topological converse to the Bers similarity principle.

Demonstrated well-posedness of the Dirichlet problem for certain non-elliptic conductivity equations.

Abstract

We study Hardy classes on the disk associated to the equation $\bar\d w=\alpha\bar w$ for $\alpha\in L^r$ with $2\leq r<\infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M.~Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in $W^{1,2}$. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for $W^{1,2}_0$-functions.