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

TL;DR

This paper introduces local Orlicz-slice Hardy spaces, characterizes their duals, and clarifies their relationships, providing insights into bilinear decompositions and the sharpness of certain embeddings.

Contribution

It defines and studies local Orlicz-slice Hardy spaces, characterizes their dual spaces, and clarifies the inclusion relations among these spaces, advancing the understanding of bilinear decompositions.

Findings

Established dual space characterizations via atoms and maximal functions.

Proved the strict inclusion $h_ullet^ ext{Phi}( e^n) eq h^{ ext{log}}( e^n)$.

Clarified the sharpness of bilinear decomposition embeddings.

Abstract

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.