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We study the Zaremba problem, or mixed problem associated to the Laplace operator, in two-dimensional Lipschitz graph domains with mixed Dirichlet and Neumann boundary data in Lebesgue and Lorentz spaces. We obtain an explicit value $r$such that the Zaremba problem is solvable in $L^p$for $1>p>r$and in the Lorentz space $L^{r,1}$. Applications include those where the domain is a cone with vertex at the origin and, more generally, a Schwarz–Christoffel domain. The techniques employed are based on results of the Zaremba problem in the upper half-plane, the use of conformal maps and the theory of solutions to the Neumann problem. For the case when the domain is the upper half-plane, we work in the weighted setting, establishing conditions on the weights for the existence of solutions and estimates for the non-tangential maximal function of the gradient of the solution. In particular, in the $L^2$-unweighted case, where known examples show that the gradient of the solution may fail to be squared-integrable, we prove restricted weak-type estimates. 

We prove Hilbert transform identities involving conformal maps via the use of Rellich identity and the solution of the Neumann problem in a graph Lipschitz domain in the plane. We obtain as consequences new $L^2$-weighted estimates for the Hilbert transform, including a sharp bound for its norm as a bounded operator in weighted $L^2$ in terms of a weight constant associated to the Helson-Szeg\"o theorem. 

We study the existence, uniqueness, and optimal regularity of solutions to trans-mission problems for harmonic functions with C1,α interfaces. For this, we develop a novel geometric stability argument based on the mean value property.