  This paper studies the regularity problem for block uniformly elliptic
operators in divergence form with complex bounded measurable coefficients. We
consider the case where the boundary data belongs to Lebesgue spaces with
weights in the Muckenhoupt classes. Our results generalize those of S.
Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order
method) who considered the unweighted case. To obtain our main results we use
the weighted Hardy space theory associated with elliptic operators recently
developed by the last two named authors. One of the novel contributions of this
paper is the use of an "inhomogeneous" vertical square function which is shown
to be controlled by the gradient of the function to which is applied in
weighted Lebesgue spaces.
