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

Abstract

Abstract:

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.

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

CLC Number:

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].Acta mathematica scientia,Series B, 2013, 33(5): 1329-1346.

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