Pseudo-holomorphic functions at the critical exponent

Abstract

We study Hardy classes on the disk associated to the equation $\bar{\partial }w=\alpha \bar{w}$ for $\alpha \in L^r$ with $2\le r<\infty$. 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 2D 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 $W_0^{1,2}$-functions.