On the boundedness of B-maximal commutators, commutators of B-Riesz potentials and B-singular integral operators in modified B-Morrey spaces

Abstract

In this paper we consider the generalized shift operator associated to the Laplace–Bessel differential operator ∆B and investigate B-maximal commutators, commutators of B-Riesz potentials and commutators of B-singular integral operators associated to the generalized shift operator. The boundedness of the B-maximal commutator Mb,γ and the commutator [b, Aγ] of the B-singular integral operator on the modified B-Morrey spaces Lep,λ,γ(R n k,+) for all 1 < p < ∞ when b ∈ BMOγ(R n k,+) are proved. In addition, we obtain that the commutator [b, Iα,γ] of the B-Riesz potential Iα,γ is bounded from the modified B-Morrey space Lep,λ,γ(R n k,+) to Leq,λ,γ(R n k,+), 1 < p < n+|γ|−λ n+|γ| ≤ 1 p − 1 q ≤ n+|γ|−λ and from the space Le1,λ,γ(R n k,+) to WLeq,λ,γ(R n k,+), n+|γ| ≤ 1 − 1 q ≤ n+|γ|−λ