  Recently, both the bilinear decompositions $h^1(\mathbb{R}^n)\times
\mathrm{\,bmo}(\mathbb{R}^n) \subset L^1
(\mathbb{R}^n)+h_\ast^\Phi(\mathbb{R}^n)$ and $h^1(\mathbb{R}^n) \times
\mathrm{bmo}(\mathbb{R}^n) \subset L^1 (\mathbb{R}^n) + h^{\log}(\mathbb{R}^n)$
were established. In this article, 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 the variant
$h_\ast^\Phi(\mathbb{R}^n)$ of the local Orlicz Hardy space introduced by A.
Bonami and J. 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 $h_\ast^{\Phi}(\mathbb R^n)$. The
relationships $h_\ast^\Phi(\mathbb{R}^n) \subsetneqq h^{\log}(\mathbb{R}^n)$ is
also clarified.
