Acta mathematica scientia, Series B >
BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS
Received date: 2011-11-30
Revised date: 2013-03-02
Online 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).
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(φ1, φ2) which ensures the boundedness of the operator Mα from one generalized Morrey space Mp, φ1 (G) to another Mq, φ2 (G), 1 < p ≤ q < ∞, 1/p − 1/q = /Q, and from the space M1, φ1 (G) to the weak space W Mq, 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 Mq, φ1/q(G) for 1 < p < q < ∞and from M1, 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 Mb,α ,k from Mp, φ1 (G) to Mq, φ2 (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 p, φ1/p(G) to Mq, 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 −ΔG + 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 Vγ (−ΔG + V )− and V γ∇G(−ΔG + V )− are obtained.
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 . DOI: 10.1016/S0252-9602(13)60085-5
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