scholarly journals Riesz Transform Characterization of Weighted Hardy Spaces Associated with Schrödinger Operators

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Hua Zhu
Keyword(s):  
Schrödinger Operator ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Nonnegative Function ◽  
Riesz Transforms ◽  
Schrodinger Operator ◽  
Weighted Hardy Spaces ◽  
Local Hardy Spaces ◽  
Reverse Hölder

We characterize the weighted local Hardy spaceshρ1(ω)related to the critical radius functionρand weightsω∈A1ρ,∞(Rn)by localized Riesz transformsR^j; in addition, we give a characterization of weighted Hardy spacesHL1(ω)via Riesz transforms associated with Schrödinger operatorL, whereL=-Δ+Vis a Schrödinger operator onRn(n≥3) andVis a nonnegative function satisfying the reverse Hölder inequality.

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.

scholarly journals Local Riesz transforms characterization of local Hardy spaces

2008 ◽  
Vol 59 (3) ◽  
pp. 299-320 ◽  
Author(s):  
Marco M. Peloso ◽  
Silvia Secco
Keyword(s):  
Hardy Spaces ◽  
Riesz Transforms ◽  
Local Hardy Spaces

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.

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.

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).

Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

2015 ◽  
Vol 67 (5) ◽  
pp. 1161-1200 ◽  
Author(s):  
Junqiang Zhang ◽  
Jun Cao ◽  
Renjin Jiang ◽  
Dachun Yang
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic Operator ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Function Characterization

AbstractLet w be either in the Muckenhoupt class of A2(ℝn) weights or in the class of QC(ℝn) weights, and let be the degenerate elliptic operator on the Euclidean space ℝn, n ≥ 2. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space associated with , and when with , the authors prove that the associated Riesz transform is bounded from to the weighted classical Hardy space .

Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schr ödinger Operators

2015 ◽  
Vol 58 (2) ◽  
pp. 432-448 ◽  
Author(s):  
Dachun Yang ◽  
Sibei Yang
Keyword(s):  
Hardy Space ◽  
Orlicz Space ◽  
Orlicz Function ◽  
Second Order ◽  
Riesz Transforms ◽  
Muckenhoupt Weights ◽  
Schrodinger Operator ◽  
Magnetic Schrödinger Operator ◽  
Maximal Inequalities ◽  
Reverse Hölder

AbstractLet be a magnetic Schrödinger operator on ℝn, wheresatisfy some reverse Hölder conditions. Let be such that ϕ(x, ·) for any given x ∊ ℝn is an Orlicz function, ϕ( ·, t) ∊ A∞(ℝn) for all t ∊ (0,∞) (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index . In this article, the authors prove that second-order Riesz transforms VA-1 and are bounded from the Musielak–Orlicz–Hardy space Hµ,A(Rn), associated with A, to theMusielak–Orlicz space Lµ(Rn). Moreover, we establish the boundedness of VA-1 on . As applications, some maximal inequalities associated with A in the scale of Hµ,A(Rn) are obtained

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