scholarly journals Commutators of Higher Order Riesz Transform Associated with Schrödinger Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Yu Liu ◽  
Lijuan Wang ◽  
Jianfeng Dong
Keyword(s):  
Hardy Space ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Higher Order ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Reverse Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator onℝn(n≥3), whereV≢0is a nonnegative potential belonging to certain reverse Hölder classBsfors≥n/2. In this paper, we prove the boundedness of commutatorsℛbHf=bℛHf-ℛH(bf)generated by the higher order Riesz transformℛH=∇2(-Δ+V)-1, whereb∈BMOθ(ρ), which is larger than the spaceBMO(ℝn). Moreover, we prove thatℛbHis bounded from the Hardy spaceHL1(ℝn)into weakLweak1(ℝn).

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$n>4$.

scholarly journals The Boundedness of Marcinkiewicz Integrals Associated with Schrödinger Operator on Morrey Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Dongxiang Chen ◽  
Fangting Jin
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Marcinkiewicz Integrals ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator, whereVbelongs to some reverse Hölder class. The authors establish the boundedness of Marcinkiewicz integrals associated with Schrödinger operators and their commutators on Morrey spaces.

scholarly journals The boundedness of commutator of Riesz transform associated with Schrödinger operators on a Hardy space

2009 ◽  
Vol 7 (3) ◽  
pp. 241-250
Author(s):  
Canqin Tang ◽  
Chuanmei Bi
Keyword(s):  
Hardy Space ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Type Function

In this paper, we study the boundedness of commutator[b,T]of Riesz transform associated with Schrödinger operator andbisBMOtype function, note that the kernel ofThas no smoothness, and the boundedness fromHb1(Rn)→L1(Rn)is obtained.

scholarly journals Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Hua Wang
Keyword(s):  
Schrödinger Operator ◽  
Morrey Spaces ◽  
Fractional Integrals ◽  
Laplacian Operator ◽  
Schrodinger Operator ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Reverse Hölder

Let ℒ=−Δ+V be a Schrödinger operator on ℝd, d≥3, where Δ is the Laplacian operator on ℝd, and the nonnegative potential V belongs to the reverse Hölder class RHs with s≥d/2. For given 0<α<d, the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐα=ℒ−α/2. Suppose that b is a locally integrable function on ℝd and the commutator generated by b and ℐα is defined by b.ℐαfx=bx⋅ℐαfx−ℐαbfx. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHs with s≥d/2. Then, we will establish the boundedness properties of the fractional integrals ℐα on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator b,ℐα in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than Ap,q, BMOℝd, and Lp,κμ,ν corresponding to the classical case (that is V≡0).

scholarly journals Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators

2019 ◽  
Vol 9 (1) ◽  
pp. 1291-1314
Author(s):  
Jizheng Huang ◽  
Pengtao Li ◽  
Yu Liu
Keyword(s):  
Hardy Space ◽  
Schrödinger Operators ◽  
Heat Semigroup ◽  
Schrodinger Operators ◽  
Reverse Hölder Class ◽  
Area Integral ◽  
Muckenhoupt Class ◽  
Degenerate Schrödinger Operator ◽  
Reverse Hölder

Abstract Let $$\begin{array}{} \displaystyle Lf(x)=-\frac{1}{\omega(x)}\sum_{i,j}^{}\partial_{i}(a_{ij}(\cdot)\partial_{j}f)(x)+V(x)f(x) \end{array}$$ be the degenerate Schrödinger operator, where ω is a weight from the Muckenhoupt class A2, V is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure ω(x)dx. For such an operator we define the area integral $\begin{array}{} \displaystyle S^{L}_h \end{array}$ associated with the heat semigroup and obtain the area integral characterization of $\begin{array}{} \displaystyle H^{1}_{L} \end{array}$, which is the Hardy space associated with L.

scholarly journals Variation inequalities related to Schrödinger operators on weighted Morrey spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 813-827
Author(s):  
Jing Zhang
Keyword(s):  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Variation Operators ◽  
Reverse Hölder

Abstract This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

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.

BMO-type estimates of Riesz transforms associated with generalized Schrödinger operators

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Yu Liu ◽  
Shuai Qi
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Generalized Schrödinger Operator ◽  
Type Decomposition

Abstract In this paper we show that the dual Riesz transform associated with the generalized Schrödinger operator {\mathcal{L}} is bounded from {\mathrm{BMO}} into {\mathrm{BMO}_{\mathcal{L}}} and give the Fefferman–Stein-type decomposition of {\mathrm{BMO}_{\mathcal{L}}} functions in terms of Riesz transforms.

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