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

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

Vagif GULIYEV1,2|Ali AKBULUT1|Yagub MAMMADOV3   

  1. 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 GULIYEV1,2|Ali AKBULUT1|Yagub MAMMADOV3   

  1. 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:

In the article we consider the fractional maximal operator Mα , 0 ≤α < Q on any Carnot group G (i.e., nilpotent stratified Lie group) in the generalized Morrey spaces Mp, φ(G), where Q is the homogeneous dimension of G. We find the conditions on the pair(φ1φ2) which ensures the boundedness of the operator Mα from one generalized Morrey space Mpφ1 (G) to another Mqφ2 (G), 1 < p q < ∞, 1/p − 1/q = /Q, and from the space M1, φ1 (G) to the weak space W Mqφ2/ (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 Mq, φ1/q(G) for 1 < p < q < ∞and from M1, φ(G) to W M qφ1/q(G) for 1 < q < ∞. In the case b ∈ BMO(G) and 1 < p < q < ∞, find the sufficient conditions on the pair (φ1φ2) which ensures the boundedness of the kth-order commutator operator Mb,α ,k from Mpφ1 (G) to Mqφ2 (G) with 1/p−1/q =α /Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb, α ,k from M pφ1/p(G) to Mqφ1/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 (φ1φ2) and φ, which do not assume any assumption on monotonicity of (φ1φ2) and φ in r. As applications we consider the SchrÖdinger operator −ΔG + V on G, where the nonnegative potential V belongs to the reverse HÖlder class B(G). The Mpφ1Mqφ2 estimates for the operators Vγ (−ΔG + V )− and V γG(−ΔG + V )− are obtained.

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

Abstract:

In the article we consider the fractional maximal operator Mα , 0 ≤α < Q on any Carnot group G (i.e., nilpotent stratified Lie group) in the generalized Morrey spaces Mp, φ(G), where Q is the homogeneous dimension of G. We find the conditions on the pair(φ1φ2) which ensures the boundedness of the operator Mα from one generalized Morrey space Mpφ1 (G) to another Mqφ2 (G), 1 < p q < ∞, 1/p − 1/q = /Q, and from the space M1, φ1 (G) to the weak space W Mqφ2/ (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 Mq, φ1/q(G) for 1 < p < q < ∞and from M1, φ(G) to W M qφ1/q(G) for 1 < q < ∞. In the case b ∈ BMO(G) and 1 < p < q < ∞, find the sufficient conditions on the pair (φ1φ2) which ensures the boundedness of the kth-order commutator operator Mb,α ,k from Mpφ1 (G) to Mqφ2 (G) with 1/p−1/q =α /Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb, α ,k from M pφ1/p(G) to Mqφ1/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 (φ1φ2) and φ, which do not assume any assumption on monotonicity of (φ1φ2) and φ in r. As applications we consider the SchrÖdinger operator −ΔG + V on G, where the nonnegative potential V belongs to the reverse HÖlder class B(G). The Mpφ1Mqφ2 estimates for the operators Vγ (−ΔG + V )− and V γG(−ΔG + V )− are obtained.

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

中图分类号: 

  • 42B25