A Remark on Estimates for Uniformly Elliptic Operators on WeightedLp Spaces and Morrey Spaces

Kurata, Kazuhiro; Sugano, Satoko

doi:10.1002/(sici)1522-2616(200001)209:1<137::aid-mana137>3.0.co;2-3

Cited by 40 publications

(27 citation statements)

References 14 publications

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“…Shen [38] studied the Schrödinger operator −Δ + V , assuming the nonnegative potential V belongs to the reverse Hölder class B q (R n ) for q ≥ n/2 and he proved the L p boundedness of the operators (−Δ+V ) iγ , ∇ 2 (−Δ+V ) −1 , ∇(−Δ+V ) − 1 2 and ∇(−Δ + V ) −1 . Kurata and Sugano generalized Shen's results to uniformly elliptic operators in [24]. Sugano [43] also extended some results of Shen to the operator…”

Section: Schrödinger Type Operatorsmentioning

confidence: 99%

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

Guliyev

Aliyev

Karaman

et al. 2011

Integr. Equ. Oper. Theory

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In this paper the authors study the boundedness for a large class of sublinear operators Tα, α ∈ [0, n) generated by Calderón-Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces Mp,ϕ. As an application of the above result, the boundeness of the commutator of sublinear operators T b,α , α ∈ [0, n) on generalized Morrey spaces is also obtained. In the case b ∈ BM O and T b,α is a sublinear operator, we find the sufficient conditions on the pair (ϕ1, ϕ2) which ensures the boundedness of the operators T b,α , α ∈ [0, n) from one generalized Morrey space Mp,ϕ 1 to another Mq,ϕ 2 with 1/p − 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (ϕ1, ϕ2), which do not assume any assumption on monotonicity of ϕ1, ϕ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.Mathematics Subject Classification (2010). Primary 42B20, 42B25, 42B35.

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Section: Schrödinger Type Operatorsmentioning

confidence: 99%

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

Guliyev

Aliyev

Karaman

et al. 2011

Integr. Equ. Oper. Theory

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“…Kurata and Sugano generalized Shens results to uniformly elliptic operators in Kurata and Sugano (2000). Sugano (1998) also extended some results of Shen to the operator…”

Section: Parabolic Schrödinger Type Operatorsmentioning

confidence: 97%

Anisotropic Fractional Maximal Operator in Anisotropic Generalized Morrey Spaces

Dzhabrailov¹,

Khaligova²

2012

JMR

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In this paper it is proved that anisotropic fractional maximal operator M α,σ , 0 ≤ α < |σ| is bounded on anisotropic generalized Morrey spaces M p,ϕ,σ , where |σ| = n i=1 σ i is the homogeneous dimension of R n . We find the conditions on the pair (ϕ 1 , ϕ 2 ) which ensure the Spanne-Guliyev type boundedness of the operator M α,σ from anisotropic generalized Morrey space M p,ϕ 1 ,σ to M q,ϕ 2 ,σ , 1 < p ≤ q < ∞, 1/p − 1/q = α/|σ|, and from the space M 1,ϕ 1 ,σ to the weak space W M q,ϕ 2 ,σ , 1 < q < ∞, 1 − 1/q = α/|σ|. We also find conditions on the ϕ which ensure the AdamsGuliyev type boundedness ofAs applications, we establish the boundedness of some Schödinger type operators on anisotropic generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

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“…(9.1) without t coordinate, see also [29]), and Okazawa [28] gave a L p (R d ) estimate for Schrödinger-type operators with nonnegative potentials V, which satisfy the condition |∇V| ≤ c p V 3/2 . In addition, Kurata and Sugano [27] studied the L p,ω (R d ) and L p,β (R d ) boundedness for uniformly elliptic operators L = (a i j (x)u x i (x)) x j with nonnegative potentials belonging to a certain elliptic-type reverse Hölder class B q (q > 1), where a i j ∈ C α with α ∈ (0, 1] and ω belong to a certain class of Muckenhoupt weights.…”

Section: Schrödinger-type Operatorsmentioning

confidence: 99%

Weighted Solvability for Parabolic Equations With Partially Bmo Coefficients and Its Applications

Lin

2014

J. Aust. Math. Soc.

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We consider the weighted L p solvability for divergence and nondivergence form parabolic equations with partially bounded mean oscillation (BMO) coefficients and certain positive potentials. As an application, global regularity in Morrey spaces for divergence form parabolic operators with partially BMO coefficients on a bounded domain is established.2010 Mathematics subject classification: primary 35J55; secondary 42B25.

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A Remark on Estimates for Uniformly Elliptic Operators on WeightedLp Spaces and Morrey Spaces

Cited by 40 publications

References 14 publications

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

Anisotropic Fractional Maximal Operator in Anisotropic Generalized Morrey Spaces

Weighted Solvability for Parabolic Equations With Partially Bmo Coefficients and Its Applications

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