Ali Akbulut

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

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 Ma from one generalized Morrey space M p , φ 1 ( G ) to another M q , φ 2 ( G ) , 1 p ≤ q ∞ , 1 / p − 1 / q = α / Q , and from the space M p , φ 1 ( G ) to the weak space W M q , φ 2 ( G ) , 1 ≤ q ∞ , 1 − 1 / q = α / Q . Also find conditions on the φ which ensure the Adams type boundedness of the Ma from M p , φ 1 p ( G ) to M p , φ 1 q ( G ) for 1 < p < q < ∞ and from M 1 , φ ( G ) to W M q , φ 1 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, a, k from M p , φ 1 ( G ) to M p , φ 2 ( G ) with 1/p − 1/q = a/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb, α, k from M p , φ 1 p ( G ) to M q , φ 1 q ( G ) for 1 < p < q < ∞. In all the cases the conditions for the boundedness of Ma are given it terms of supremal-type 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 Schrodinger operator − Δ G + V on G , where the nonnegative potential V belongs to the reverse Holder class B∞( G ). The Mp, φ1 − Mq, φ2 estimates for the operators V γ ( − Δ G + V ) − β and V γ ∇ G ( − Δ G + V ) − β are obtained.

Boundedness of the anisotropic Riesz potential in anisotropic local Morrey-type spaces

The problem of boundedness of the anisotropic Riesz potential in local Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.