scholarly journals Second order Riesz transforms associated to the Schrödinger operator forp⩽1

2014 ◽  
Vol 410 (1) ◽  
pp. 391-402 ◽  
Author(s):  
Fu Ken Ly
Keyword(s):  
Schrödinger Operator ◽  
Second Order ◽  
Riesz Transforms ◽  
Schrodinger Operator

scholarly journals Regularity for the second order Riesz transforms associated with Schrödinger operators on weighted BMO type spaces

2018 ◽  
Vol 20 ◽  
pp. 02005
Author(s):  
Trong Nguyen Ngoc ◽  
Dao Nguyen Anh ◽  
L. X. Truong
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Hölder’S Inequality ◽  
Second Order ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Weighted Bmo ◽  
Type Spaces ◽  
Weighted Case

Let L = −Δ + V be a Schrödinger operator on ℝn, where V is a nonnegative potential satisfying the suitable reverse Hölder’s inequality. In this paper, we study the boundedness of the second order Riesz transforms such as L−1∇2 on the spaces of BMO type for weighted case. We generalized the known results to the weighted case.

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 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.

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