Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2021; 58(1): 235-251

Online first article November 3, 2020      Printed January 31, 2021

https://doi.org/10.4134/BKMS.b200240

Copyright © The Korean Mathematical Society.

Estimates for the higher order Riesz transforms related to Schr\"odinger type operators

Yanhui Wang

Jiaozuo University

Abstract

We consider the Schr\"odinger type operator \(\mathcal{L}_k=(-\Delta)^k+V^k \) on \(\mathbb{R}^n ( n\geq 2k+1)\), where \(k=1,2\) and the nonnegative potential \(V\) belongs to the reverse H\"older class \(RH_s\) with \( n/21/2,$ where $p_1=\frac{n}{4(\beta-\alpha)-2}$, $p_2=\frac{n}{n-4(\beta-\alpha)+2}.$ Moreover, we prove that $T_{\alpha,\beta}$ is bounded from $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$ to itself when $\beta-\alpha=1/2.$

Keywords: Riesz transform, Schr\"odinger operator, Hardy space, BMO

MSC numbers: Primary 35J10, 42B35