Intrinsic Structures of Certain Musielak-Orlicz Hardy Spaces

For any $p\in (0,1]$, let $H^{{\mathrm{\Phi }}_p}({\mathbb{R}}^n)$ be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function ${\mathrm{\Phi }}_p$, defined by setting, for any $x\in {\mathbb{R}}^n$ and $t\in [0,\mathrm{\infty })$, $${\mathrm{\Phi }}_p(x,t):=\{\begin{array}{ll}\frac{t}{\log (e+t)+[t(1+\mathrm{\mid }x\mathrm{\mid }{)}^n{]}^{1-p}}& \text{when }n(1\mathrm{/}p-1)\mathrm{\notin }\mathbb{N}\cup \{0\};\\ \frac{t}{\log (e+t)+[t(1+\mathrm{\mid }x\mathrm{\mid }{)}^n{]}^{1-p}[\log (e+\mathrm{\mid }x\mathrm{\mid }){]}^p}& \text{when }n(1\mathrm{/}p-1)\in \mathbb{N}\cup \{0\},\end{array}$$ 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, $H^{{\mathrm{\Phi }}_1}({\mathbb{R}}^n)$ 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 $H^{{\mathrm{\Phi }}_p}({\mathbb{R}}^n)$ by showing that, for any $p\in (0,1]$, $H^{{\mathrm{\Phi }}_p}({\mathbb{R}}^n)=H^{{\varphi }_0}({\mathbb{R}}^n)+H_{W_p}^p({\mathbb{R}}^n)$ 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, $H^{{\varphi }_0}({\mathbb{R}}^n)$ the Orlicz-Hardy space associated with the Orlicz function ${\varphi }_0(t):=t\mathrm{/}\log (e+t)$ for any $t\in [0,\mathrm{\infty })$ and $H_{W_p}^p({\mathbb{R}}^n)$ the weighted Hardy space associated with certain weight function $W_p(x)$ that is comparable to ${\mathrm{\Phi }}_p(x,1)$ 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.