  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.
