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.

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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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