Behaviour of Schr\"odinger Riesz transforms over smoothness spaces

TL;DR

This paper investigates the boundedness of Schr"odinger Riesz transforms on smoothness spaces, identifying minimal potential conditions needed for boundedness on weighted BMO spaces, extending understanding beyond classical $L^p$ bounds.

Contribution

It introduces new minimal conditions on potentials to ensure boundedness of Schr"odinger Riesz transforms on weighted BMO spaces, expanding the regularity analysis of these operators.

Findings

Boundedness of Riesz transforms on weighted BMO spaces under new potential conditions

Identification of minimal additional conditions necessary for boundedness

Extension of regularity results for Schr"odinger operators

Abstract

As it was shown by Shen, the Riesz transforms associated to the Schr\"odinger operator $L=-\Delta + V$ are not bounded on $L^p(\mathbb{R}^d)$-spaces for all $p, 1<p<\infty$, under the only assumption that the potential satisfies a reverse H\"older condition of order $d/2$, $d\geq3$. Furthermore, they are bounded only for $p$ in some finite interval of the type $(1,p_0)$, so it can not be expected to preserve regularity spaces. In this work we search for some kind of minimal additional conditions on the potential in order to obtain boundedness on appropriate weighted $BMO$ type regularity spaces for all first and second order Riesz transforms, namely for the operators $\nabla L^{-1/2}$, $V^{1/2}L^{-1/2}$, $\nabla^2 L^{-1}$, $VL^{-1}$ and $V^{1/2}\nabla L^{-1}$. We also explore to what extent such extra conditions are also necessary.