Intrinsic Structures of Certain Musielak–Orlicz Hardy Spaces

Abstract

For any $$p\in (0,\,1]$$ , let $$H^{\Phi _p}(\mathbb {R}^n)$$ be the Musielak–Orlicz Hardy space associated with the Musielak–Orlicz growth function $$\Phi _p$$ , defined by setting, for any $$x\in \mathbb {R}^n$$ and $$t\in [0,\,\infty )$$ , $$\begin{aligned}&\Phi _{p}(x,\,t)\\&\quad := {\left\{ \begin{array}{ll} \displaystyle \frac{t}{\log {(e+t)}+[t(1+|x|)^n]^{1-p}}&{} \quad \text {when} n(1/p-1)\notin \mathbb N \cup \displaystyle \frac{t}{\log (e+t)+[t(1+|x|)^n]^{1-p}[\log (e+|x|)]^p}&{} \quad \text {when} n(1/p-1)\in \mathbb N\cup \{0\}, \end{array}\right. } \end{aligned}$$ which is the sharp target space of the bilinear decomposition of the product of the Hardy space $$H^p(\mathbb {R}^n)$$ and its dual. Moreover, $$H^{\Phi _1}(\mathbb {R}^n)$$ is the prototype appearing in the real-variable theory of general Musielak–Orlicz Hardy spaces. In this article, the authors find a new structure of the space $$H^{\Phi _p}(\mathbb {R}^n)$$ by showing that, for any $$p\in (0,\,1]$$ , $$H^{\Phi _p}(\mathbb {R}^n)=H^{\phi _0}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})$$ and, for any $$p\in (0,\,1)$$ , $$H^{\Phi _p}(\mathbb {R}^n)=H^{1}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})$$ , where $$H^1(\mathbb {R}^n)$$ denotes the classical real Hardy space, $$H^{\phi _0}({{{\mathbb {R}}}^n})$$ the Orlicz–Hardy space associated with the Orlicz function $$\phi _0(t):=t/\log (e+t)$$ for any $$t\in [0,\infty )$$ , and $$H_{W_p}^p(\mathbb {R}^n)$$ the weighted Hardy space associated with certain weight function $$W_p(x)$$ that is comparable to $$\Phi _p(x,1)$$ for any $$x\in \mathbb {R}^n$$ . As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.