Parameter Marcinkiewicz integral and its commutator on generalized Orlicz-Morrey spaces

Abstract

. The aim of this paper is to mainly establish the sufficient and necessary conditions for the boundedness of the commutator M ρ Ω ,b which is generated by the parameter Marcinkiwicz integral M ρ Ω and the Lipschitz function b on generalized Orlicz-Morrey space L Φ ,ϕ ( R d ) in the sense of the Adams type result (or Spanne type result). Moreover, the necessary conditions for the parameter Marcinkiewizcz integral M ρ Ω on the L Φ ,ϕ ( R d ), and the commutator [ b, M ρ Ω ] generated by the M ρ Ω and the space BMO on the L Φ ,ϕ ( R d ), are also obtained, respectively.