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 End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yanping Chen ◽  
Wenyu Tao
Keyword(s):  
Schrödinger Operator ◽  
Radon Measure ◽  
Hölder Inequality ◽  
Point Estimate ◽  
Schrodinger Operator ◽  
Reverse Hölder Inequality ◽  
Scale Invariant ◽  
End Point ◽  
Reverse Hölder ◽  
Generalized Schrödinger Operator

Let L = − Δ + μ be the generalized Schrödinger operator on ℝ d , d ≥ 3 , where μ ≠ 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new BMO space associated to the generalized Schrödinger operator L , BM O θ , L , which is bigger than the BMO spaces related to the classical Schrödinger operators A = − Δ + V , with V a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to L in BM O θ , L also be proved.

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 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.

RIESZ TRANSFORMS AND LITTLEWOOD–PALEY SQUARE FUNCTION ASSOCIATED TO SCHRÖDINGER OPERATORS ON NEW WEIGHTED SPACES

2018 ◽  
Vol 105 (2) ◽  
pp. 201-228
Author(s):  
NGUYEN NGOC TRONG ◽  
LE XUAN TRUONG
Keyword(s):  
Schrödinger Operators ◽  
Hölder Inequality ◽  
Riesz Transforms ◽  
Bounded Mean Oscillation ◽  
Schrodinger Operators ◽  
Square Function ◽  
Reverse Hölder Inequality ◽  
Lipschitz Spaces ◽  
Mean Oscillation ◽  
Reverse Hölder

Let ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+{\mathcal{V}}$ be a Schrödinger operator on $\mathbb{R}^{n},n\geq 3$, where ${\mathcal{V}}$ is a potential satisfying an appropriate reverse Hölder inequality. In this paper, we prove the boundedness of the Riesz transforms and the Littlewood–Paley square function associated with Schrödinger operators ${\mathcal{L}}$ in some new function spaces, such as new weighted Bounded Mean Oscillation (BMO) and weighted Lipschitz spaces, associated with ${\mathcal{L}}$. Our results extend certain well-known results.

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

The reverse Hölder inequality for power means

2012 ◽  
Vol 183 (6) ◽  
pp. 762-771
Author(s):  
Viktor D. Didenko ◽  
Anatolii A. Korenovskyi
Keyword(s):  
Hölder Inequality ◽  
Reverse Hölder Inequality ◽  
Power Means ◽  
Reverse Hölder
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