On weak$^*$-convergence in $H^1_L(\mathbb R^d)$

TL;DR

This paper extends the classical weak$^*$-convergence theorem to the Hardy space associated with a Schr"odinger operator on $\,\mathbb{R}^d$, under specific conditions on the potential V.

Contribution

It establishes a version of Jones and Journé's theorem for weak$^*$-convergence in $H^1_L(\mathbb{R}^d)$ involving Schr"odinger operators with potentials in $RH_{d/2}$.

Findings

Proves weak$^*$-convergence in $H^1_L(\mathbb{R}^d)$ for Schr"odinger operators.

Extends classical harmonic analysis results to Schr"odinger Hardy spaces.

Provides foundational results for analysis involving Schr"odinger operators.

Abstract

Let $L= -\Delta+ V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative function, $V\ne 0$, and belongs to the reverse H\"older class $RH_{d/2}$. In this paper, we prove a version of the classical theorem of Jones and Journ\'e on weak$^*$-convergence in the Hardy space $H^1_L(\mathbb R^d)$.