Arkiv för Matematik

Volume 60 (2022)

Number 2

Notes on $H^{\log}$: structural properties, dyadic variants, and bilinear $H^1$-$\operatorname{BMO}$ mappings

Pages: 231 – 275

DOI: https://dx.doi.org/10.4310/ARKIV.2022.v60.n2.a2

Authors

Odysseas Bakas (Centre for Mathematical Sciences, Lund University, Lund, Sweden; and BCAM-Basque Center for Applied Mathematics, Bilbao, Spain)

Sandra Pott (Centre for Mathematical Sciences, Lund University, Lund, Sweden)

Salvador Rodríguez-López (Department of Mathematics, Stockholm University, Stockholm, Sweden)

Alan Sola (Department of Mathematics, Stockholm University, Stockholm, Sweden)

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.

Keywords

maximal function, real Hardy spaces, Orlicz spaces, Haar wavelets

2010 Mathematics Subject Classification

Primary 42B35. Secondary 42B25, 42C40.

Full Text (PDF format)

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.

Received 4 December 2020

Received revised 9 September 2021

Accepted 20 September 2021

Published 26 October 2022