BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

doi:10.1016/S0252-9602(13)60085-5

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (5): 1329-1346.doi: 10.1016/S0252-9602(13)60085-5

BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

Vagif GULIYEV^1,2|Ali AKBULUT¹|Yagub MAMMADOV³

1. Department of Mathematics, Ahi Evran University, 40100, Kirsehir, Turkey;
2. Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku;
3. Nakhchivan Teacher-Training Institute, Nakchivan, Azerbaijan

收稿日期:2011-11-30 修回日期:2013-03-02 出版日期:2013-09-20 发布日期:2013-09-20
基金资助:
The research of V. Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Projects (FEN 4001.12.0018) and by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1. V. Guliyev and A. Akbulut were partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695). The research of A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Projects (FEN 4001.12.0019).

BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

Vagif GULIYEV^1,2|Ali AKBULUT¹|Yagub MAMMADOV³

1. Department of Mathematics, Ahi Evran University, 40100, Kirsehir, Turkey;
2. Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku;
3. Nakhchivan Teacher-Training Institute, Nakchivan, Azerbaijan

Received:2011-11-30 Revised:2013-03-02 Online:2013-09-20 Published:2013-09-20
Supported by:
The research of V. Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Projects (FEN 4001.12.0018) and by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1. V. Guliyev and A. Akbulut were partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695). The research of A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Projects (FEN 4001.12.0019).

摘要/Abstract

摘要：

In the article we consider the fractional maximal operator M_α , 0 ≤α < Q on any Carnot group G (i.e., nilpotent stratified Lie group) in the generalized Morrey spaces M_p, φ(G), where Q is the homogeneous dimension of G. We find the conditions on the pair(φ₁, φ₂) which ensures the boundedness of the operator M_α from one generalized Morrey space M_p_, φ1 (G) to another M_q_, φ2 (G), 1 < p ≤ q < ∞, 1/p − 1/q = /Q, and from the space M_1, φ1 (G) to the weak space W M_q_, φ2/ (G), 1 ≤ q < ∞, 1 − 1/q =α /Q. Also find conditions on the φ which ensure the Adams type boundedness of the M from M
_p_, φ1/p(G) to M_q, φ_1/q(G) for 1 < p < q < ∞and from M_1, φ(G) to W M _q, φ1/q(G) for 1 < q < ∞. In the case b ∈ BMO(G) and 1 < p < q < ∞, find the sufficient conditions on the pair (φ₁, φ₂) which ensures the boundedness of the kth-order commutator operator M_b_{,α ,k} from M_p_, φ1 (G) to M_q_, φ2 (G) with 1/p−1/q =α /Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator M_{b, α ,k} from M _p_, φ1/p(G) to M_q_, φ1/q(G) for 1 < p < q < ∞. In all the cases the conditions for the boundedness of M are given it terms of supremaltype inequalities on (φ₁, φ₂) and φ, which do not assume any assumption on monotonicity of (φ₁, φ₂) and φ in r. As applications we consider the SchrÖdinger operator −Δ_G + V on G, where the nonnegative potential V belongs to the reverse HÖlder class B_∞(G). The Mp, φ₁ −Mq, φ₂ estimates for the operators V^γ (−Δ_G + V )− and V^γ∇_G(−Δ_G + V )− are obtained.

关键词: Carnot group, fractional maximal function, generalized Morrey space, SchrÖdinger operator, BMO space

Abstract:

Key words: Carnot group, fractional maximal function, generalized Morrey space, SchrÖdinger operator, BMO space

中图分类号:

42B25

Vagif GULIYEV, Ali AKBULUT, Yagub MAMMADOV. BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS[J]. 数学物理学报(英文版), 2013, 33(5): 1329-1346.

Vagif GULIYEV, Ali AKBULUT, Yagub MAMMADOV. BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS[J]. Acta mathematica scientia,Series B, 2013, 33(5): 1329-1346.

参考文献

[1] Adams D R. A note on Riesz potentials. Duke Math, 1975, 42: 765–778
[2] Akbulut A, Guliyev V S, Mustafayev R. On the Boundedness of the maximal operator and singular integral
operators in generalized Morrey spaces. Math Bohem, 2012, 137(1): 27–43
[3] Alphonse A M. An end point estimate for maximal commutators. J Fourier Anal Appl, 2000, 6(4): 449–456
[4] Burenkov V, Gogatishvili A, Guliyev V S, Mustafayev R. Boundedness of the fractional maximal operator
in local Morrey-type spaces. Complex Var Elliptic Equ, 2010, 55(8–10): 739–758
[5] Fefferman C. The uncertainty principle. Bull Amer Math Soc, 1983, 9: 129–206
[6] Folland G B. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark Mat, 1975, 13:
161–207
[7] Folland G B, Stein E M. Hardy Spaces on Homogeneous Groups. Math Notes, 28. Princeton: Princeton
Univ Press, 1982
[8] Giaquinta M. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton,
NJ: Princeton Univ Press, 1983
[9] Guliyev V S. Integral operators on function spaces on the homogeneous groups and on domains in Rn (in
Russian) [D]. Moscow: Mat Inst Steklova, 1994: 1–329
[10] Guliyev V S. Function spaces, Integral Operators and TwoWeighted Inequalities on Homogeneous Groups.
Some Applications (Russian). Baku: ELM, 1996
[11] Guliyev V S. Boundedness of the maximal, potential and singular operators in the generalized Morrey
spaces. J Inequal Appl, 2009, Art ID 503948, 20 pp.
[12] Guliyev V S, Mustafayev R. Fractional integrals in spaces of functions defined on spaces of homogeneous
type. Anal Math, 1998, 24(3): 181–200
[13] Guliyev V S, Aliyev S, Karaman T, Shukurov P. Boundedness of sublinear operators and commutators on
generalized Morrey spaces. Integral Equations Operator Theory, 2011, 71(3): 327–355
[14] Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratics
forms. Trans Amer Math Soc, 1980, 258: 147–153
[15] Kurata K, Sugano S. A remark on estimates for uniformly elliptic operators on weighted Lp spaces and
Morrey spaces. Math Nachr, 2000, 209: 137–150
[16] Li H Q. Estimations Lp des operateurs de Schr¨odinger sur les groupes nilpotents. J Funct Anal, 1999,
161: 152–218
[17] Liu Yu. The weighted estimates for the operators V (−G + V )− and V ∇G(−G + V )− on the
stratified Lie group G. J Math Anal Appl, 2009, 349: 235–244
[18] Lu G Z. A FeffermanPhong type inequality for degenerate vector fields and applications. Panamer Math
J, 1996, 6: 37–57
[19] Lu G, Lu Sh, Yang D, Singular integrals and commutators on homogeneous groups. Anal Math J, 2002,
28: 103–143
[20] Morrey C B. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc,
1938, 43: 126–166
[21] Nakai E. Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized
Morrey spaces. Math Nachr, 1994, 166: 95–103
[22] Nakai E. The Campanato, Morrey and H¨older spaces on spaces of homogeneous type. Studia Math, 2006,
176 (1): 1–19

[23] Polidoro S, Ragusa M A. Holder regularity for solutions of ultraparabolic equations in divergence form.
Potential Analysis, 2001, 14(4): 341–350
[24] Shen Z W. Lp estimates for Schr¨odinger operators with certain potentials. Ann Inst Fourier (Grenoble),
1995, 45: 513–546
[25] Stein E M. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton,
New Jersey: Princeton Univ Press, 1993
[26] Str¨omberg J O, Torchinsky A.Weighted Hardy Spaces. Lecture Notes in Math, Vol 1381. Berlin: Springer-
Verlag, 1989
[27] Sugano S. Estimates for the operators V (− + V )− and V ∇(− + V )− with certain nonnegative
potentials V . Tokyo J Math, 1998, 21: 441–452
[28] Varopoulos N, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge Univ Press,
1992
[29] Zhong J P. Harmonic analysis for some Schr¨odinger type operators [D]. Princeton University, 1993

BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

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