scholarly journals Some Estimates of Schrödinger-Type Operators with Certain Nonnegative Potentials

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Yu Liu ◽  
Youzheng Ding
Keyword(s):  
Weak Type ◽  
Type Operator ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weak Type Estimates ◽  
Holder Class ◽  
Schrödinger Type Operator ◽  
Schrödinger Type Operators ◽  
Reverse Hölder

We consider the Schrödinger-type operatorH=(−Δ)2+V2, where the nonnegative potentialVbelongs to the reverse Hölder classBq1forq1≥n/2,  n≥5. TheLpestimates of the operator∇4H−1related toHare obtained whenV∈Bq1and1<p≤q1/2. We also obtain the weak-type estimates of the operator∇4H−1under the same condition ofV.

scholarly journals Boundedness for some Schrödinger Type Operators on Weighted Morrey Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Guixia Pan ◽  
Lin Tang
Keyword(s):  
Morrey Spaces ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Schrödinger Type Operators ◽  
Reverse Hölder

We establish the boundedness of some Schrödinger type operators on weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

scholarly journals Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

2019 ◽  
Vol 2019 ◽  
pp. 1-18
Author(s):  
Hua Wang
Keyword(s):  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Strong Type ◽  
Type Estimate ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator, whereΔis the Laplacian onRdand the nonnegative potentialVbelongs to the reverse Hölder classRHqforq≥d. The Riesz transform associated with the operatorL=-Δ+Vis denoted byR=∇(-Δ+V)-1/2and the dual Riesz transform is denoted byR⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classRHqforq≥d. Then we will establish the mapping properties of the operatorRand its adjointR⁎on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[b,R]and[b,R⁎]are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanAp,BMO(Rd), andLp,κ(w)corresponding to the classical Riesz transforms (V≡0).

scholarly journals Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Vagif S. Guliyev ◽  
Kamala R. Rahimova
Keyword(s):  
Maximal Operator ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Reverse Hölder Class ◽  
Fractional Maximal Operator ◽  
Hölder Class ◽  
Holder Class ◽  
Homogeneous Dimension ◽  
Reverse Hölder ◽  
Limiting Case

We prove that the parabolic fractional maximal operatorMαP,0≤α<γ, is bounded from the modified parabolic Morrey spaceM̃1,λ,P(ℝn)to the weak modified parabolic Morrey spaceWM̃q,λ,P(ℝn)if and only ifα/γ≤1-1/q≤α/(γ-λ)and fromM̃p,λ,P(ℝn)toM̃q,λ,P(ℝn)if and only ifα/γ≤1/p-1/q≤α/(γ-λ). Hereγ=trPis the homogeneous dimension onℝn. In the limiting case(γ-λ)/α≤p≤γ/αwe prove that the operatorMαPis bounded fromM̃p,λ,P(ℝn)toL∞(ℝn). As an application, we prove the boundedness ofMαPfrom the parabolic Besov-modified Morrey spacesBM̃pθ,λs(ℝn)toBM̃qθ,λs(ℝn). As other applications, we establish the boundedness of some Schrödinger-ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

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 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 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.

scholarly journals Schrödinger Operators with Reverse Hölder Class Potentials in the Dunkl Setting and Their Hardy Spaces

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Agnieszka Hejna
Keyword(s):  
Hardy Space ◽  
Root System ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Multiplicity Function ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

AbstractFor a normalized root system R in $${\mathbb {R}}^N$$ R N and a multiplicity function $$k\ge 0$$ k ≥ 0 let $${\mathbf {N}}=N+\sum _{\alpha \in R} k(\alpha )$$ N = N + ∑ α ∈ R k ( α ) . We denote by $$dw({\mathbf {x}})=\varPi _{\alpha \in R}|\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}\,d{\mathbf {x}}$$ d w ( x ) = Π α ∈ R | ⟨ x , α ⟩ | k ( α ) d x the associated measure in $${\mathbb {R}}^N$$ R N . Let $$L=-\varDelta +V$$ L = - Δ + V , $$V\ge 0$$ V ≥ 0 , be the Dunkl–Schrödinger operator on $${\mathbb {R}}^N$$ R N . Assume that there exists $$q >\max (1,\frac{{\mathbf {N}}}{2})$$ q > max ( 1 , N 2 ) such that V belongs to the reverse Hölder class $$\mathrm{{RH}}^{q}(dw)$$ RH q ( d w ) . We prove the Fefferman–Phong inequality for L. As an application, we conclude that the Hardy space $$H^1_{L}$$ H L 1 , which is originally defined by means of the maximal function associated with the semigroup $$e^{-tL}$$ e - t L , admits an atomic decomposition with local atoms in the sense of Goldberg, where their localizations are adapted to V.

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$n>4$.

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