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 \(-\textrm{div}A \nabla \) $-\text{div}A\mathrm{\nabla }$ 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 \(L^2\) $L^2$ 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- \(L^p\) $L^p$ “ \(N<S\) $N<S$ ” estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full \(L^2\) $L^2$ 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 \(L^p\) $L^p$ -solvability of boundary value problems for the magnetic Schrödinger operator \(-(\nabla -i\textbf{a})^2+V\) $-{(\mathrm{\nabla }-i\mathbf{a})}^2+V$ when the magnetic potential \(\textbf{a}\) $\mathbf{a}$ and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.