Some estimates for commutators of Littlewood-Paley g-functions

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 gg-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 Lp(μ){L}^{p}\left(\mu ) into Lebesgue spaces Lp(μ){L}^{p}\left(\mu ) for p∈(1,∞)p\in \left(1,\infty ) and also bounded from spaces L1(μ){L}^{1}\left(\mu ) into spaces L1,∞(μ){L}^{1,\infty }\left(\mu ). Furthermore, the boundedness of [b,g˙rb,{\dot{g}}_{r}] on Morrey space Mqp(μ){M}_{q}^{p}\left(\mu ) and on generalized Morrey Lp,ϕ(μ){L}^{p,\phi }\left(\mu ) is obtained.

Keywords

• non-homogeneous metric measure space
• littlewood-paley g-function
• commutator
• space rbmo(μ)
• generalized morrey space
• 42b25
• 42b35
• 30l99