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

Laboratory of Mathematics and Complex Systems, Ministry of Education of China, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China

E-mail Address: yangyzhang@mail.bnu.edu.cn

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Dachun Yang

Laboratory of Mathematics and Complex Systems, Ministry of Education of China, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China

E-mail Address: dcyang@bnu.edu.cn

Corresponding author.

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, and

Wen Yuan

Laboratory of Mathematics and Complex Systems, Ministry of Education of China, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China

E-mail Address: wenyuan@bnu.edu.cn

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https://doi.org/10.1142/S0219199721500048Cited by:15 (Source: Crossref)

Abstract

Recently, both the bilinear decompositions $h^{1} (ℝ^{n}) \times bmo (ℝ^{n}) \subset L^{1} (ℝ^{n}) + h_{*}^{Φ} (ℝ^{n})$ and $h^{1} (ℝ^{n}) \times bmo (ℝ^{n}) \subset L^{1} (ℝ^{n}) + h^{log} (ℝ^{n})$ 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 $h_{*}^{Φ} (ℝ^{n})$ , 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 $h_{*}^{Φ} (ℝ^{n})$ . The relationship $h_{*}^{Φ} (ℝ^{n}) ⊊ h^{log} (ℝ^{n})$ is also clarified.

Keywords:

AMSC: 42B30, 42B15, 42B35, 46E30

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