Weighted sharp function inequalities and boundedness for commutator of Riesz transforms of Schrödinger operator

2014 ◽  
Vol 25 (10) ◽  
pp. 765-776 ◽  
Author(s):  
Wenxin Yu ◽  
Xiangyu Wang
Keyword(s):  
Schrödinger Operator ◽  
Riesz Transforms ◽  
Schrodinger Operator ◽  
Sharp Function

THE ESTIMATES OF RIESZ TRANSFORMS ASSOCIATED TO SCHRÖDINGER OPERATORS

2016 ◽  
Vol 101 (3) ◽  
pp. 290-309 ◽  
Author(s):  
QINGQUAN DENG ◽  
YONG DING ◽  
XIAOHUA YAO
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Reverse Hölder Class ◽  
Holder Class ◽  
Small Norm ◽  
Reverse Hölder

Let$H=-\unicode[STIX]{x1D6E5}+V$be a Schrödinger operator with some general signed potential$V$. This paper is mainly devoted to establishing the$L^{q}$-boundedness of the Riesz transform$\unicode[STIX]{x1D6FB}H^{-1/2}$for$q>2$. We mainly prove that under certain conditions on$V$, the Riesz transform$\unicode[STIX]{x1D6FB}H^{-1/2}$is bounded on$L^{q}$for all$q\in [2,p_{0})$with a given$2<p_{0}<n$. As an application, the main result can be applied to the operator$H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$, where$V_{+}$belongs to the reverse Hölder class$B_{\unicode[STIX]{x1D703}}$and$V_{-}\in L^{n/2,\infty }$with a small norm. In particular, if$V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$for some positive number$\unicode[STIX]{x1D6FE}$,$\unicode[STIX]{x1D6FB}H^{-1/2}$is bounded on$L^{q}$for all$q\in [2,n/2)$and$n>4$.

Weighted Boundedness of Commutators of Riesz Transforms Associated with Schrödinger Operator

2012 ◽  
Vol 256-259 ◽  
pp. 2939-2942
Author(s):  
Wen Hua Gao ◽  
Wei Zhou
Keyword(s):  
Schrödinger Operator ◽  
Lipschitz Function ◽  
Hölder Inequality ◽  
Riesz Transforms ◽  
Schrodinger Operator ◽  
Reverse Hölder Inequality ◽  
Lebesgue Integral ◽  
Weighted Function ◽  
Weighted Lipschitz Function ◽  
Reverse Hölder

In this paper, the Schrödinger operator on n dimensions Euclid space with the non-zero, nonnegative potential function satisfying the reverse Hölder inequality is considered. The weighted boundedness of the commutators composed of several Riesz transforms associated with the Schrödinger operator and weighted Lipschitz function on weighted Lebesgue integral spaces are obtained, for some weighted function.

Weighted BMO Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operator

2013 ◽  
Vol 303-306 ◽  
pp. 1613-1617
Author(s):  
Wen Hua Gao
Keyword(s):  
Schrödinger Operator ◽  
Hölder Inequality ◽  
Riesz Transforms ◽  
Schrodinger Operator ◽  
Reverse Hölder Inequality ◽  
Weighted Function ◽  
Bmo Function ◽  
Weighted Bmo ◽  
Bmo Spaces ◽  
Reverse Hölder

Schrödinger operator; Weighted BMO spaces; Reverse Hölder inequality; Commutator Abstract. In this paper, the Schrödinger operator on n dimensions Euclid space with the non-zero, nonnegative potential function satisfying the reverse Hölder inequality is considered. The weighted boundedness of the commutators composed of several Riesz transforms associated with the Schrödinger operator and weighted BMO function on weighted Lebesgue integral spaces are obtained, for some weighted function.

Endpoint Estimates of Riesz Transforms Associated with Generalized Schrödinger Operators

2018 ◽  
Vol 61 (4) ◽  
pp. 787-801 ◽  
Author(s):  
Yu Liu ◽  
Shuai Qi
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Hardy Type ◽  
Generalized Schrödinger Operator

AbstractIn this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized Schrödinger operator.

scholarly journals ENDPOINT ESTIMATES FOR COMMUTATORS OF RIESZ TRANSFORMS ASSOCIATED WITH SCHRÖDINGER OPERATORS

2010 ◽  
Vol 82 (3) ◽  
pp. 367-389 ◽  
Author(s):  
PENGTAO LI ◽  
LIZHONG PENG
Keyword(s):  
Hardy Space ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Schrodinger Operator

AbstractIn this paper, we discuss the H1L-boundedness of commutators of Riesz transforms associated with the Schrödinger operator L=−△+V, where H1L(Rn) is the Hardy space associated with L. We assume that V (x) is a nonzero, nonnegative potential which belongs to Bq for some q>n/2. Let T1=V (x)(−△+V )−1, T2=V1/2(−△+V )−1/2 and T3 =∇(−△+V )−1/2. We prove that, for b∈BMO (Rn) , the commutator [b,T3 ] is not bounded from H1L(Rn) to L1 (Rn) as T3 itself. As an alternative, we obtain that [b,Ti ] , ( i=1,2,3 ) are of (H1L,L1weak) -boundedness.

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