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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. 

Fractional Leibniz rules are reminiscent of the product rule learned in calculus classes, offering estimates in the Lebesgue norm for fractional derivatives of a product of functions in terms of the Lebesgue norms of each function and its fractional derivatives. We prove such estimates for Coifman-Meyer multiplier operators in the setting of Triebel-Lizorkin and Besov spaces based on quasi-Banach function spaces. In particular, these include rearrangement invariant quasi-Banach function spaces such as weighted Lebesgue spaces, weighted Lorentz spaces and generalizations, and Orlicz spaces. The method used also yields results in weighted mixed Lebesgue spaces and Morrey spaces, where we present applications to the specific case of power weights, as well as in variable Lebesgue spaces. 

We present an overview of Besov and Triebel–Lizorkin spaces in the Hermite setting and applications on boundedness properties of Hermite pseudo-multipliers and fractional Leibniz rules in such spaces. We also give a new weighted estimate for Hermite multipliers for weights related to Hermite operators.