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 $t\in [0,\,\infty )$,

$$\begin{aligned}&\Phi _{p}(x,\,t)\\&\quad := {\left\{ \begin{array}{ll} \displaystyle \frac{t}{\log {(e+t)}+[t(1+|x|)^n]^{1-p}}&{} \quad \text {when}\ n(1/p-1)\notin \mathbb N \cup \{0\},\\ \displaystyle \frac{t}{\log (e+t)+[t(1+|x|)^n]^{1-p}[\log (e+|x|)]^p}&{} \quad \text {when}\ n(1/p-1)\in \mathbb N\cup \{0\}, \end{array}\right. } \end{aligned}$$

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.