Riesz transforms associated with higher-order Schr\"odinger type operators

TL;DR

This paper investigates the boundedness of Riesz transforms associated with higher-order Schrödinger operators, establishing sharp $L^q$ bounds under various regularity conditions on the potential.

Contribution

It provides new $L^q$ boundedness results for Riesz transforms linked to higher-order Schrödinger operators, including sharp bounds and conditions for different $q$ ranges.

Findings

Established $L^q$ boundedness for $q eq 2$ under subcritical potential conditions.

Derived sharp $L^q$ bounds for Riesz transforms with regularity assumptions on the potential.

Applied results to operators involving fractional Laplacians with inverse power potentials.

Abstract

In this paper, let $L=L_{0}+V$ be a Schr\"{o}dinger type operator where $L_{0}$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption on $V$, the authors study the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q\leq 2$ and obtain a sharp result. Furthermore, the authors impose extra regularity assumptions on $V$ to obtain the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q>2$. As an application, the main results can be applied to the operator $L=(-\Delta)^{m}-\gamma|x|^{-2m}$ for suitable $\gamma$