Abstract
This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy–Littlewood, Zygmund, and Stein for $ H^{\log}$ and related Musielak–Orlicz spaces.
Funding Statement
The first author was partially supported by the ‘Wallenberg Mathematics Program 2018’, grant no. KAW 2017.0425, by the Spanish Government through SEV-2017-0718, RYC2018- 025477-I, PID2021-122156NB-I00 / AEI / 10.13039/501100011033 funded by Agencia Estatal de Investigación and acronym ‘HAMIP’, Juan de la Cierva Incorporación IJC2020-043082-I, and by the Basque Government through BERC 2022-2025. The second author was partially supported by VR grant 2015-05552. The third author was partially supported by the Spanish Government grant PID2020-113048GB-I00.
Citation
Odysseas Bakas. Sandra Pott. Salvador Rodríguez-López. Alan Sola. "Notes on $H^{\log}$: structural properties, dyadic variants, and bilinear $H^1$-$\operatorname{BMO}$ mappings." Ark. Mat. 60 (2) 231 - 275, October 2022. https://doi.org/10.4310/ARKIV.2022.v60.n2.a2
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