Square functions associated to Schrodinger operators

TL;DR

This paper characterizes Banach space geometry through the boundedness of Schrödinger operator-associated square functions, offering new proofs for classical harmonic analysis results.

Contribution

It introduces a refined localization method to analyze square functions linked to Schrödinger operators with reverse Hölder potentials, providing alternative proofs for key boundedness properties.

Findings

Characterization of Banach space geometry via square functions

Refined localization method for Schrödinger operators

Alternative proofs for boundedness in H^1, L^p, and BMO

Abstract

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrodinger operators of the form $L=-\Delta+V$, where the nonnegative potential $V$ satisfies a reverse Holder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in $H^1$, $L^p$ and $BMO$ of classical $L$-square functions.