Pseudo-holomorphic functions at the critical exponent

Source 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 \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 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^{1,2}_0 -functions.