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This article is devoted to a study of the Hardy space $H^{\log}(\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present analternative approach to their result relating the product of a function in thereal 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 ofclassical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log}$ andrelated Musielak-Orlicz spaces.

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