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

TL;DR

This work analyzes boundary-value problems for singular Schrödinger equations of the form $-\mathrm{div}(A\nabla u)+aVu=0$ in the upper half-space with $t$-independent block-structured coefficients and a nonnegative potential $V$ in a reverse Hölder class. It develops an operator-adapted framework built on Riesz-transform bounds, Dziubanski–Zienkiewicz Hardy spaces $\textup{H}^{p}_{V}$ and Hardy–Sobolev spaces, and a holomorphic functional-calculus approach to connect second-order Schrödinger operators with first-order systems via $BD$ and $DB$. The main contributions include sharp extrapolation intervals for Dirichlet, Regularity, and Neumann problems around $p=2$, identifications of $\mathbb{H}^{p}_{H}$ and $\mathbb{H}^{1,p}_{H}$ with concrete function spaces, and new atom/molecule decompositions that accommodate rough potentials. The results yield well-posed boundary-value problems for a broad range of $L^{p}$ and Hardy-type data, with comparability of square-function and non-tangential maximal-function controls for corresponding solutions, thereby offering a robust, operator-adapted PDE framework for equations with block-structured, non-symmetric coefficients and singular potentials.

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 $n\geq 3$. 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 Hölder 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 Dziubanski--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.

Figures (1)

• Figure 1: Relations between various adapted Hardy spaces

Theorems & Definitions (160)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof