Real-variable characterizations of local Orlicz-slice Hardy spaces with application to bilinear decompositions

Recently, both the bilinear decompositions h(1)(R-n) x bmo(R-n) subset of L-1(R-n) + h(*)(Phi)(R-n) and h1(R-n) x bmo(R-n) subset of 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(*)(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(*)(Phi)(R-n). The relationship h(*)(Phi)(R-n) (subset of)(not equal) h(log)(R-n) is also clarified.

Automated summary

This paper explores two cases of the bilinear decompositions and proves that one case is sharp while the other is not. It also introduces the local Orlicz-slice Hardy space and investigates its properties, providing further insights into its characteristics.