Abstract. We establish optimal L p bounds for the non-tangential maximal function of the gradient of the solution to a second-order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the vertical variable, on the domain above a Lipschitz graph in the plane, in terms of the L p -norm at the boundary of the tangential derivative of the Dirichlet data, or of the Neumann data.

We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander S m ρ,δ classes. These results are new in the case ρ < 1, that is, outwith the scope of multilinear Calderón-Zygmund theory.

Let Ω be a Lipschitz domain in ℝn,n≥2, and L=div A∇ be a second‐order elliptic operator in divergence form. We establish the solvability of the Dirichlet regularity problem with boundary data in H1,p(∂Ω) and of the Neumann problem with Lp(∂Ω) data for the operator L on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator L to be rough, obeying a certain Carleson condition with small norm. These results complete the results of Dindoš, Petermichl, and Pipher (2007), where the H1,p(∂Ω) Dirichlet problem was considered under the same assumptions, and Dindoš and Rule (2010), where the regularity and Neumann problems were considered on two‐dimensional domains.© 2016 Wiley Periodicals, Inc.

Abstract. In this paper we prove weighted norm inequalities with weights in the Ap classes, for pseudodifferential operators with symbols in the class S n(ρ−1) ρ,δ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield L p boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in OPS m ρ,δ .