Bulletin of the
Korean Mathematical Society
BKMS

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

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2019; 56(5): 1117-1127

Online first article August 2, 2019      Printed September 30, 2019

https://doi.org/10.4134/BKMS.b180836

Copyright © The Korean Mathematical Society.

Estimates for Riesz transforms associated with Schr\"odinger type operators

Yueshan Wang

Jiaozuo University

Abstract

Let \(\mathcal{L}_2=(-\Delta)^2+V^2 \) be the Schr\"odinger type operator, where nonnegative potential \(V\) belongs to the reverse H\"older class $RH_s$, $s> n/2$. In this paper, we consider the operator $T_{\alpha,\beta}=V^{2\alpha} \mathcal{L}_2^{-\beta}$ and its conjugate $T^*_{\alpha,\beta}$, where $0<\alpha\leq \beta\leq 1$. We establish the $(L^{p},L^{q})$-boundedness of operator $T_{\alpha,\beta}$ and $T^*_{\alpha,\beta}$, respectively, we also show that $T_{\alpha,\beta}$ is bounded from Hardy type space \(H^1_{\mathcal{L}_2}(\mathbb{R}^n)\) into $L^{p_2}(\mathbb{R}^n)$ and $T^*_{\alpha,\beta}$ is bounded from $L^{p_1}(\mathbb{R}^n)$ into $BMO$ type space $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$, where $p_1=\frac{n}{4(\beta-\alpha)}$, $p_2=\frac{n}{n-4(\beta-\alpha)}$.

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

MSC numbers: Primary 42B35, 35J10