Some estimates for commutators of Littlewood-Paley g-functions

Abstract

Abstract The aim of this paper is to establish the boundedness of commutator [ b , g ˙ r ] \\left[b,{\\dot{g}}_{r}] generated by Littlewood-Paley g g -functions g ˙ r {\\dot{g}}_{r} and b ∈ RBMO ( μ ) b\\in {\\rm{RBMO}}\\left(\\mu ) on non-homogeneous metric measure space. Under assumption that λ \\lambda satisfies ε \\varepsilon -weak reverse doubling condition, the author proves that [ b , g ˙ r ] \\left[b,{\\dot{g}}_{r}] is bounded from Lebesgue spaces L p ( μ ) {L}^{p}\\left(\\mu ) into Lebesgue spaces L p ( μ ) {L}^{p}\\left(\\mu ) for p ∈ ( 1 , ∞ ) p\\in \\left(1,\\infty ) and also bounded from spaces L 1 ( μ ) {L}^{1}\\left(\\mu ) into spaces L 1 , ∞ ( μ ) {L}^{1,\\infty }\\left(\\mu ) . Furthermore, the boundedness of [ b , g ˙ r b,{\\dot{g}}_{r} ] on Morrey space M q p ( μ ) {M}_{q}^{p}\\left(\\mu ) and on generalized Morrey L p , ϕ ( μ ) {L}^{p,\\phi }\\left(\\mu ) is obtained.