Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

Abstract

Let $\vec a: = ({a_1}, \ldots ,{a_n}) \in {[1,\infty )^n},\,p \in (0,1)$, and α:= 1/p − 1. For any x ∈ ℝⁿ and t ∈ [0, ∞), let

$${\Phi _p}(x,t): = \left\{ {\matrix{{{t \over {1 + {{(t[x]_{\vec a}^\nu )}^{1 - p}}}}} \hfill & {{\rm{if}}\,\nu \alpha \notin \mathbb{N},} \hfill \cr {{t \over {1 + {{(t[x]_{\vec a}^\nu )}^{1 - p}}{{[\log ({\rm{e}} + |x{|_{\vec a}})]}^p}}}} \hfill & {{\rm{if}}\,\nu \alpha \in \mathbb{N},} \hfill \cr } } \right.$$

where ${[ \cdot ]_{\vec a}}: = 1 + | \cdot {|_{\vec a}},\,| \cdot {|_{\vec a}}$ denotes the anisotropic quasi-homogeneous norm with respect to ${\vec a}$, and ν:= a₁ +...+ a_n. Let $H_{\vec a}^p({\mathbb{R}^n})$, ${\cal L}_\alpha ^{\vec a}({\mathbb{R}^n})$, and $H_{\vec a}^{{\Phi _p}}({\mathbb{R}^n})$ be, respectively, the anisotropic Hardy space, the anisotropic Campanato space, and the anisotropic Musielak-Orlicz Hardy space associated with Φ_p on ℝⁿ. In this article, via first establishing the wavelet characterization of anisotropic Campanato spaces, we prove that for any $f \in H_{\vec a}^p({\mathbb{R}^n})$ and $g \in {\cal L}_\alpha ^{\vec a}({\mathbb{R}^n})$, the product of f and g can be decomposed into S(f, g) + T(f, g) in the sense of tempered distributions, where S is a bilinear operator bounded from $H_{\vec a}^p({\mathbb{R}^n}) \times {\cal L}_\alpha ^{\vec a}({\mathbb{R}^n})$ to L¹(ℝⁿ) and T is a bilinear operator bounded from $H_{\vec a}^p({\mathbb{R}^n}) \times {\cal L}_\alpha ^{\vec a}({\mathbb{R}^n})$ to $H_{\vec a}^{{\Phi _p}}({\mathbb{R}^n})$. Moreover, this bilinear decomposition is sharp in the dual sense that any ${\cal Y} \subset H_{\vec a}^{{\Phi _p}}({\mathbb{R}^n})$ that fits into the above bilinear decomposition should satisfy ${({L^1}({\mathbb{R}^n}) + {\cal Y})^ * } = {({L^1}({\mathbb{R}^n}) + H_{\vec a}^{{\Phi _p}}({\mathbb{R}^n}))^ * }$. As applications, for any non-constant $b \in {\cal L}_\alpha ^{\vec a}({\mathbb{R}^n})$ and any sublinear operator T satisfying some mild bounded assumptions, we find the largest subspace of $H_{\vec a}^p({\mathbb{R}^n})$, denoted by $H_{\vec a,b}^p({\mathbb{R}^n})$, such that the commutator [b, T] is bounded from $H_{\vec a,b}^p({\mathbb{R}^n})$ to L¹(ℝⁿ). In addition, when T is an anisotropic Calderón-Zygmund operator, the boundedness of [b, T] from $H_{\vec a,b}^p({\mathbb{R}^n})$ to L¹(ℝⁿ)(or to $H_{\vec a}^1({\mathbb{R}^n})$) is also presented. The key of their proofs is the wavelet characterization of function spaces under consideration.