Pseudo-holomorphic functions at the critical exponent

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.