  We study boundary value problems for degenerate elliptic equations and
systems with square integrable boundary data. We can allow for degeneracies in
the form of an $A_{2}$ weight. We obtain representations and boundary traces
for solutions in appropriate classes, perturbation results for solvability and
solvability in some situations. The technology of earlier works of the first
two authors can be adapted to the weighted setting once the needed quadratic
estimate is established and we even improve some results in the unweighted
setting. The proof of this quadratic estimate does not follow from earlier
results on the topic and is the core of the article.
