Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
References
All Topics
Mathematics
Pure Mathematics

Commutators of Marcinkiewicz integrals associated with Schrödinger operator on generalized weighted Morrey spaces

Profile image of Vagif S. GuliyevVagif S. Guliyev

2016, Journal of Mathematical Inequalities

https://doi.org/10.7153/JMI-10-77
visibility

…

description

24 pages

Abstract

Let L = −Δ + V be a Schrödinger operator, where Δ is the Laplacian on R n , while nonnegative potential V belongs to the reverse Hölder class. Let also Ω ∈ L q (S n−1) be a homogeneous function of degree zero with q > 1 and have a mean value zero on S n−1. In this paper, we study the boundedness of the Marcinkiewicz operators μ L j,Ω and their commutators μ L j,Ω,b with rough kernels associated with Schrödinger operator on generalized weighted Morrey spaces M p,ϕ (w). We find the sufficient conditions on the pair (ϕ 1 ,ϕ 2) with q < p < ∞ and w ∈ A p/q or 1 < p < q and w 1−p ∈ A p /q which ensures the boundedness of the operators μ L j,Ω from one generalized weighted Morrey space M p,ϕ 1 (w) to another M p,ϕ 2 (w) for 1 < p < ∞. We find the sufficient conditions on the pair (ϕ 1 ,ϕ 2) with b ∈ BMO(R n) and q < p < ∞ , w ∈ A p/q or 1 < p < q , w 1−p ∈ A p /q which ensures the boundedness of the operators μ L j,Ω,b , j = 1,... ,n from M p,ϕ 1 (w) to M p,ϕ 2 (w) for 1 < p < ∞. In all cases the conditions for the boundedness of the operators μ L j,Ω , μ L j,Ω,b , j = 1,... ,n are given in terms of Zygmund-type integral inequalities on (ϕ 1 ,ϕ 2) and w , which do not assume any assumption on monotonicity of ϕ 1 (x,r), ϕ 2 (x,r) in r .

Related papers

The boundedness of certain sublinear operators with rough kernel generated by Calder\'on-Zygmund operators and their commutators on generalized weighted Morrey spaces
FERİT GÜRBÜZ

arXiv (Cornell University), 2016

The aim of this paper is to get the boundedness of certain sublinear operators with rough kernel generated by Calderón-Zygmund operators on the generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutator operators formed by BM O functions and certain sublinear operators with rough kernel are also bounded on the generalized weighted Morrey spaces. Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

View PDFchevron_right
Boundedness of Sublinear Operators Generated by Calderón-Zygmund Operators on Generalized Weighted Morrey Spaces
Vagif S. Guliyev

Annals of the Alexandru Ioan Cuza University - Mathematics, 2014

In this paper we study the boundedness for a large class of sublinear operators T generated by Calderón-Zygmund operators on generalized weighted Morrey spaces Mp,φ(w) with the weight function w(x) belonging to Muckenhoupt's class Ap. We find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the operator T from one generalized weighted Morrey space Mp,φ 1 (w) to another Mp,φ 2 (w) for p > 1 and from M1,φ 1 (w) to the weak space W M1,φ 2 (w). In all 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 pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

View PDFchevron_right
Weighted Hardy and potential operators in the generalized Morrey spaces
Natasha Samko

Journal of Mathematical Analysis and Applications, 2011

We study the weighted p → q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces L p,ϕ (R n , w) defined by an almost increasing function ϕ(r) and radial type weight w(|x|). We obtain sufficient conditions, in terms of some integral inequalities imposed on ϕ and w, for such a p → q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p → q-boundedness of the Riesz potential operator.

View PDFchevron_right
Boundedness of some operators on weighted amalgam spaces
Songbai Wang

2021

Let t ∈ (0,∞), p ∈ (1,∞), q ∈ [1,∞], w ∈ Ap and v ∈ Aq. We introduce the weighted amalgam space (L, L)t(R ) and show some properties of it. Some estimates on these spaces for the classical operators in harmonic analysis, such as the Hardy–Littlewood maximal operator, the Calderón–Zygmund operator, the Riesz potential, singular integral operators with the rough kernel, the Marcinkiewicz integral, the Bochner-Riesz operator, the Littlewood-Paley g function and the intrinsic square function, are considered. Our main method is extrapolation. We obtain some new weak results for these operators on weighted amalgam spaces.

View PDFchevron_right
On the boundedness of generalized Cesàro operators on Sobolev spaces
carlos lizama

Journal of Mathematical Analysis and Applications, 2014

View PDFchevron_right
Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces
Vagif S. Guliyev

Journal of Mathematical Inequalities, 2011

We prove that the fractional maximal operator M α and the Riesz potential operator I α , 0 < α < n are bounded from the modified Morrey space L 1,λ (R n) to the weak modified Morrey space W L q,λ (R n) if and only if, α/n 1 − 1/q α/(n − λ) and from L p,λ (R n) to L q,λ (R n) if and only if, α/n 1/p − 1/q α/(n − λ). As applications, we establish the boundedness of some Schödinger type operators on modified Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on modified Morrey spaces which are estimated by Riesz potentials.

View PDFchevron_right
Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I
Victor Burenkov

The survey is aimed at providing detailed information about recent results in the problem of the boundedness in general Morrey-type spaces of various important operators of real analysis, namely of the maximal operator, fractional maximal operator, Riesz potential, singular integral operator, Hardy operator. The main focus is on the results which contain, for a certain range of the numerical parameters, necessary and sufficient conditions on the functional parameters characterizing general Morrey-type spaces, ensuring the boundedness of the aforementioned operators from one general Morrey-type space to another one. The major part of the survey is dedicated to the results obtained by the author jointly with his co-authores A.

View PDFchevron_right
On some subspaces of Morrey–Sobolev spaces and boundedness of Riesz integrals
Ibrahim Fofana

Annales Polonici Mathematici, 2013

For 1 ≤ q ≤ α ≤ p ≤ ∞, (L q , l p) α is a complex Banach space which is continuously included in the Wiener amalgam space (L q , l p) and contains the Lebesgue space L α. We study the closure (L q , l p) α c,0 in (L q , l p) α of the space D of test functions (infinitely differentiable and with compact support in R d) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space W 1 ((L q , l p) α) (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space W 1,α) and obtain in it Sobolev inequalities and a Kondrashov-Rellich compactness theorem.

View PDFchevron_right
Oscillatory Integral Operators and Their Commutators in Modified Weighted Morrey Spaces with Variable Exponent
Javanshir Hasanov

International Journal of Applied Mathematics, 2019

In this paper first we prove Calderón-Zygmund-type integral inequalities for oscillatory integral operators and their commutators in the modified weighted Morrey spaces with variable exponent L p(•),λ ω (Ω), where Ω ⊂ R n are unbounded sets. After that we prove the boundedness of these operators on the spaces L p(•),λ ω (Ω).

View PDFchevron_right
Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces
Vagif S. Guliyev

Abstract and Applied Analysis, 2011

The authors study the boundedness for a large class of sublinear operatorTgenerated by Calderón-Zygmund operator on generalized Morrey spacesMp,φ. As an application of this result, the boundedness of the commutator of sublinear operatorsTaon generalized Morrey spaces is obtained. In the casea∈BMO(ℝn),1&lt;p&lt;∞andTais a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operatorTafrom one generalized Morrey spaceMp,φ1to anotherMp,φ2. In all cases, the conditions for the boundedness ofTaare given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity ofφ1,φ2inr. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

View PDFchevron_right

References (44)

  1. A. AKBULUT, V. S. GULIYEV AND R. MUSTAFAYEV, On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces, Math. Bohem. 137 (1) (2012), 27-43.
  2. A. AKBULUT, O. KUZU, Marcinkiewicz integrals with rough kernel associated with Schrödinger operator on vanishing generalized Morrey spaces, Azerb. J. Math. 4 (1) (2014), 40-54.
  3. J. ALVAREZ, R. J. BAGBY, D. S. KURTZ AND C. P ÉREZ, Weighted estimates for commutators of linear operators, Studia Math. 104 (1993), 195-209.
  4. A. BENEDEK, A. P. CALDERON, R. PANZONE, Convolution operators on Banach value functions, Proc. Natl. Acad. Sci. USA 48 (1962), 256-265.
  5. A. P. CALDERON, Commutators of singular integral operators, Proc. Natl. Acad. Sci. USA 53 (1965), 1092-1099.
  6. A. P. CALDERON, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA 74 (4) (1977), 1324-1327.
  7. F. CHIARENZA, M. FRASCA, Morrey spaces and Hardy-Littlewood maximal function, Rend Mat. 7 (1987), 273-279.
  8. F. CHIARENZA, M. FRASCA, P. LONGO, W 2,p -solvability of Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853.
  9. R. COIFMAN, R. ROCHBERG, G. WEISS, Factorization theorems for Hardy spaces in several vari- ables, Ann. of Math. 103 (2) (1976), 611-635.
  10. Y. CHEN, Regularity of solutions to elliptic equations with VMO coefficients, Acta Math. Sin. (Engl. Ser.) 20 (2004), 1103-1118.
  11. Y. DING, On Marcinkiewicz integral, in: Proc. of the Conference "Singular Integrals and Related Topics, III," Osaka, Japan, 2001, pp. 28-38.
  12. Y. DING, D. FAN, Y. PAN, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J. 48 (1999), 1037-1055.
  13. Y. DING, D. FAN, Y. PAN, L p -boundedness of Marcinkiewicz integrals with Hardy space function kernels, Acta Math. Sinica, English Series, 16 (2000), 593-600.
  14. Y. DING, S. LU, K. YABUTA, On commutators of Marcinkiewicz integrals with rough kernel, J. Math. Anal. Appl, 275 (2002), 60-68.
  15. J. DUOANDIKOETXEA, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869-880.
  16. J. DZIUBA ŃSKI, J. ZIENKIEWICZ, Hardy space H 1 associated to Schrödinger operator with poten- tial satisfying reverse Hölder inequality, Rev. Mat. Iber. 15 (1999), 279-296.
  17. G. DI FAZIO, M. A. RAGUSA, Interior estimates in Morrey spaces for strong solutions to nondiver- gence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 241-256.
  18. D. FAN, S. LU AND D. YANG, Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica (N. S.) 14 (1998), suppl., 625-634.
  19. W. GAO, L. TANG, Boundedness for Marcinkiewicz integrals associated with Schrödinger operators, Proc. Indian Acad. Sci. Math. Sci. 124 (2) (2014), 193-203.
  20. M. GIAQUINTA, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Prince- ton Univ. Press, Princeton, NJ, 1983.
  21. V. S. GULIYEV, Integral operators on function spaces on the homogeneous groups and on domains in R n , Doctoral dissertation, Moscow, Mat. Inst. Steklov, 1994, 329 pp. (in Russian).
  22. V. S. GULIYEV, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. 2009, Art. ID 503948, 20 pp.
  23. V. S. GULIYEV, Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J. 3 (3) (2012), 33-61.
  24. V. S. GULIYEV, S. S. ALIYEV, T. KARAMAN, Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces, Abstr. Appl. Anal. vol. 2011, Art. ID 356041, 18 pp. doi:10.1155/2011/356041
  25. V. S. GULIYEV, S. S. ALIYEV, Boundedness of parametric Marcinkiewicz integral operator and their commutators on generalized Morrey spaces, Georgian Math. J. 19 (2012), 195-208.
  26. V. S. GULIYEV, Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math. 3 (2) (2013), 79-94.
  27. V. S. GULIYEV, Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N. Y.) 193 (2) (2013), 211-227.
  28. V. S. GULIYEV, T. KARAMAN, R. CH. MUSTAFAYEV AND A. SERBETCI, Commutators of sub- linear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces, Czechoslovak Math. J. 64 (139) (2) (2014), 365-386.
  29. V. S. GULIYEV, M. N. OMAROVA, Higher order commutators of vector-valued intrinsic square func- tions on vector-valued generalized weighted Morrey spaces, Azerb. J. Math. 4 (2) (2014), 64-85.
  30. L. H ÖRMANDER, Translation invariant operators, Acta Math. 104 (1960), 93-139.
  31. S. JANSON, On functions with conditions on the mean oscillation, Ark. Mat. 14 (2) (1976), 189-196.
  32. T. KARAMAN, V. S. GULIYEV, A. SERBETCI, Boundedness of sublinear operators generated by Calderón-Zygmund operators on generalized weighted Morrey spaces, Scientic Annals of "Al. I. Cuza" University of Iasi, 60 (1) 2014, 227-244.
  33. Y. KOMORI AND S. SHIRAI, Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2) (2009), 219-231.
  34. A. KUFNER, O. JOHN AND S. FUC ¸IK, Function Spaces, Noordhoff International Publishing: Leyden, Publishing House Czechoslovak Academy of Sciences: Prague, 1977.
  35. G. LU, S. LU, D. YANG, Singular integrals and commutators on homogeneous groups, Anal. Math. 28 (2002), 103-134.
  36. T. MIZUHARA, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer-Verlag, Tokyo (1991), 183-189.
  37. C. B. MORREY, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166.
  38. B. MUCKENHOUPT, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.
  39. Z. SHEN, L p estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Greno- ble) 45 (1995), 513-546.
  40. F. SORIA, G. WEISS, A remark on singular integrals and power weights, Indiana Univ. Math. J. 43 (1994), 187-204.
  41. E. M. STEIN, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466.
  42. T. WALSH, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203-217.
  43. H. XU, J. C. CHEN AND Y. M. YING, A note on Marcinkiewicz integrals with H 1 kernels, Acta. Math. Scientia, Ser. B., 23 (1) (2003) 133-138.
  44. Received November 28, 2014) Vagif S. Guliyev Ahi Evran University, Department of Mathematics 40100, Kirsehir, Turkey and Institute of Mathematics and Mechanics of NASA AZ 1141 Baku, Azerbaijan Ali Akbulut Ahi Evran University, Department of Mathematics 40100, Kirsehir, Turkey e-mail: [email protected] Vugar H. Hamzayev nstitute of Mathematics and Mechanics of NASA AZ 1141 Baku, Azerbaijan and Nakhchivan Teacher-Training Institute, Nakhchivan, Azerbaijan e-mail: [email protected] Okan Kuzu Ahi Evran University, Department of Mathematics 40100, Kirsehir, Turkey e-mail: [email protected]
About the author
Vagif S. Guliyev
Papers
238
Followers
29
View all papers from Vagif S. Guliyevarrow_forward

Related papers

Marcinkiewicz integrals associated with Schrödinger operator on generalized Morrey spaces
OKAN KUZU

Journal of Mathematical Inequalities, 2014

Let L =-+ V be a Schrödinger operator, where is the Laplacian on R n and the non-negative potential V belongs to the reverse Hölder class RH q for q ≥ n/2. In this paper, we study the boundedness of the Marcinkiewicz integral operators μ L j and their commutators [b, μ L j ] with b ∈ BMO θ (ρ) on generalized Morrey spaces M α,V p,ϕ (R n) associated with Schrödinger operator and vanishing generalized Morrey spaces VM α,V p,ϕ (R n) associated with Schrödinger operator. We find the sufficient conditions on the pair (ϕ 1 , ϕ 2) which ensure the boundedness of the operators μ L j from one vanishing generalized Morrey space VM α,V p,ϕ 1 to another VM α,V p,ϕ 2 , 1 < p < ∞ and from the space VM α,V 1,ϕ 1 to the weak space VWM α,V 1,ϕ 2. When b belongs to BMO θ (ρ) and (ϕ 1 , ϕ 2) satisfies some conditions, we also show that [b, μ L j ] is bounded from M α,V p,ϕ 1 to M α,V p,ϕ 2 and from VM α,V p,ϕ 1 to VM α,V p,ϕ 2 , 1 < p < ∞. MSC: 42B35; 35J10

View PDFchevron_right
Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators
Ferit Gürbüz

Chinese Annals of Mathematics, Series B

Let L = −∆+V (x) be a Schrödinger operator, where ∆ is the Laplacian on R n , while nonnegative potential V (x) belonging to the reverse Hölder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrödinger operator and their commutators. Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schrödinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schrödinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood (H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.

View PDFchevron_right
Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators
Kazuhiro Kurata

Proceedings of the American Mathematical Society, 1999

In this paper, we study boundedness of integral operators on generalized Morrey spaces and its application to estimates in Morrey spaces for the Schrödinger operator L 2 = − Δ + V ( x ) + W ( x ) L_2=-\Delta +V(x)+W(x) with nonnegative V ∈ ( R H ) ∞ V\in (RH)_{\infty } (reverse Hölder class) and small perturbed potentials W W .

View PDFchevron_right
Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces
Vagif S. Guliyev

Journal of Inequalities and Applications, 2009

We consider generalized Morrey spaces M p,ω R n with a general function ω x, r defining the Morrey-type norm. We find the conditions on the pair ω 1 , ω 2 which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space M p,ω1 R n to another M p,ω2 R n , 1 < p < ∞, and from the space M 1,ω1 R n to the weak space WM 1,ω2 R n. We also prove a Sobolev-Adams type M p,ω1 R n → M q,ω2 R n-theorem for the potential operators I α. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω 1 , ω 2 , which do not assume any assumption on monotonicity of ω 1 , ω 2 in r. As applications, we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.

View PDFchevron_right
BMO estimates on vanishing generalized Morrey spaces for commutators of Marcinkiewicz integrals with rough kernel associated with schr\"odinger operator
FERİT GÜRBÜZ

arXiv (Cornell University), 2016

Let L = −∆ + V (x) be a Schrödinger operator, where ∆ is the Laplacian on R n , while nonnegative potential V (x) belonging to the reverse Hölder class. We establish the boundedness of the commutators of Marcinkiewicz integrals with rough kernel associated with schrödinger operator on vanishing generalized Morrey spaces.

View PDFchevron_right

Related topics

  • Pure Mathematics
  • Mathematical inequalities

    • Cited by

    Marcinkiewicz integrals associated with Schrödinger operator and their commutators on vanishing generalized Morrey spaces
    Vagif S. Guliyev

    Boundary Value Problems, 2017

    Let L =-+ V be a Schrödinger operator, where is the Laplacian on R n and the non-negative potential V belongs to the reverse Hölder class RH q for q ≥ n/2. In this paper, we study the boundedness of the Marcinkiewicz integral operators μ L j and their commutators [b, μ L j ] with b ∈ BMO θ (ρ) on generalized Morrey spaces M α,V p,ϕ (R n) associated with Schrödinger operator and vanishing generalized Morrey spaces VM α,V p,ϕ (R n) associated with Schrödinger operator. We find the sufficient conditions on the pair (ϕ 1 , ϕ 2) which ensure the boundedness of the operators μ L j from one vanishing generalized Morrey space VM α,V p,ϕ 1 to another VM α,V p,ϕ 2 , 1 < p < ∞ and from the space VM α,V 1,ϕ 1 to the weak space VWM α,V 1,ϕ 2. When b belongs to BMO θ (ρ) and (ϕ 1 , ϕ 2) satisfies some conditions, we also show that [b, μ L j ] is bounded from M α,V p,ϕ 1 to M α,V p,ϕ 2 and from VM α,V p,ϕ 1 to VM α,V p,ϕ 2 , 1 < p < ∞. MSC: 42B35; 35J10

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2026 Academia. All rights reserved