Boundary value problems and Hardy spaces for singular Schr\"odinger equations with block structure

TL;DR

This paper develops a framework for solving boundary value problems for singular Schrödinger equations with block-structured coefficients and singular potentials, establishing well-posedness and Hardy space characterizations in the upper half-space.

Contribution

It introduces operator-adapted Hardy spaces and Riesz transform bounds for singular Schrödinger equations with block-structured coefficients, extending solvability to boundary data in Hardy and Sobolev spaces.

Findings

Established Riesz transform bounds for the operators.

Proved well-posedness of Dirichlet and Regularity problems in L^p and Hardy spaces.

Demonstrated comparability of square functions and nontangential maximal functions.

Abstract

We obtain Riesz transform bounds and characterise operator-adapted Hardy spaces to solve boundary value problems for singular Schr\"odinger equations $-\mathrm{div}(A\nabla u)+aVu=0$ in the upper half-space $\mathbb{R}^{1+n}_{+}$ with boundary dimension $n\geq 3$. 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 H\"older class $\mathrm{RH}^{q}(\mathbb{R}^{n})$ for $q\geq \max\{\frac{n}{2},2\}$. 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 Dziuba\`{n}ski--Zienkiewicz Hardy space $\mathrm{H}^{1}_{V}(\mathbb{R}^{n})$ and adapted Hardy--Sobolev spaces $\dot{\mathrm{H}}^{1,p}_{V}(\mathbb{R}^{n})$ when $p\leq 1$. We also obtain comparability of square functions and nontangential maximal functions for the solutions with their boundary data.