Boundary value problems and Hardy spaces for singular Schrödinger equations with block structure

Abstract

We obtain Riesz transform bounds and characterise operator-adapted Hardy spaces to solve boundary value problems for singular Schrödinger equations $-\mathrm{div}(A\nabla u)+aVu=0$ in the upper half-space ${\mathbb{R}}_{+}^{1+n}$ with boundary dimension $n\ge 3$. The coefficients $(A,a,V)$ are assumed to be independent of the transversal direction to the boundary, and consist of a complex-elliptic pair $(A,a)$ that is bounded and measurable with a certain block structure, and a non-negative singular potential $V$ in the reverse Hölder class ${\mathrm{RH}}^q({\mathbb{R}}^n)$ for $q\ge \max \{\frac{n}{2},2\}$. This block structure is significant because it allows for coefficients that are not symmetric but for which ${\mathrm{L}}^2({\mathbb{R}}^n)$-solvability persists due to recently obtained Kato square root type estimates. We find extrapolation intervals for exponents $p$ around $2$ on which the Dirichlet problem is well-posed for boundary data in ${\mathrm{L}}^p({\mathbb{R}}^n)$, and the associated Regularity problem is well-posed for boundary data in Sobolev spaces ${\dot{\mathcal{V}}}^{1,p}({\mathbb{R}}^n)$ that are adapted to the potential $V$, when $p>1$. The well-posedness of these Dirichlet problems and related estimates then allow us to solve the corresponding Neumann problem with boundary data in ${\mathrm{L}}^p$. The results permit boundary data in the Dziubanski--Zienkiewicz Hardy space ${\mathrm{H}}_V^1({\mathbb{R}}^n)$ and adapted Hardy--Sobolev spaces ${\dot{\mathrm{H}}}_V^{1,p}({\mathbb{R}}^n)$ when $p\le 1$. We also obtain comparability of square functions and nontangential maximal functions for the solutions with their boundary data.

Added treatment of the Neumann problem