The Commutators of Strongly Singular Integral Operators on the Weighted Hardy Spaces

Let T be a strongly singular Calderón-Zygmund operator and b ∈ Lloc(ℝn). This article finds out a class of non-trivial subspaces BMOω,p,n(ℝn)of BMO(ℝn) for certain ω ∈ A1, 0 < p ≤ 1 and 1 < u ≤ ∞, such that the commutator [b, T] is bounded from weighted Hardy space Hωp(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\omega ^p({\mathbb{R}^n})$$\end{document} to weighted Lebesgue space Lωp(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\omega ^p({\mathbb{R}^n})$$\end{document} if b ∈ BMOω,p,∞(ℝn), and is bounded from weighted Hardy space Hωp(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\omega ^p({\mathbb{R}^n})$$\end{document} to itself if T*1 = 0 and b ∈ BMOω,p,u(ℝn)for 1 < u < 2.