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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 theformer is sharp, while the latter is not. To this end, the authors firstintroduce 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 byestablishing its characterizations via atoms, finite atoms and various maximalfunctions, which are new even for $h_\ast^{\Phi}(\mathbb R^n)$. Therelationships $h_\ast^\Phi(\mathbb{R}^n) \subsetneqq h^{\log}(\mathbb{R}^n)$ isalso clarified.

Real-Variable Characterizations of Local Orlicz-Slice Hardy Spaces with Application to Bilinear Decompositions