Recently, both the bilinear decompositions $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h^{log}({ℝ}^n)$ were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$, a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$. The relationship $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)⊊h^{log}({ℝ}^n)$ is also clarified.

最近，双线性分解$H^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset {\mathrm{大号}}^1({ℝ}^n)+H_{*}^{\mathrm{\Phi }}({ℝ}^n)$和$H^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset {\mathrm{大号}}^1({ℝ}^n)+H^{日志}({ℝ}^n)$成立。在本文中，作者在某种意义上证明了前者是尖锐的，而后者则不是。为此，作者首先引入了局部 Orlicz-slice Hardy 空间，其中包含$H_{*}^{\mathrm{\Phi }}({ℝ}^n)$, 是局部 Orlicz Hardy 空间的一个变体，由 Bonami 和 Feuto 作为特例引入，并通过原子、有限原子和各种极大函数建立其表征来获得其对偶空间，即使对于$H_{*}^{\mathrm{\Phi }}({ℝ}^n)$. 关系$H_{*}^{\mathrm{\Phi }}({ℝ}^n)⊊H^{日志}({ℝ}^n)$也得到了澄清。