Critical Perturbations for Second Order Elliptic Operators—Part II: Non-tangential Maximal Function Estimates

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators -divA∇ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the L² well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi–Nash–Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-L^p “N<S” estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full L² bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. As a corollary, we claim the first results in an unbounded domain concerning the L^p-solvability of boundary value problems for the magnetic Schrödinger operator -(∇-ia)²+V when the magnetic potential a and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.