Pseudo-holomorphic functions at the critical exponent

Abstract

We study Hardy classes on the disk associated to the equa- tion ¯ @w = � ¯ w for � 2 L r with 2 � r < 1. 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 prin- ciple. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth do- mains 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.