Sandra Pott: A dyadic approach to Hardy–Orlicz spaces and a path to the multiparameter setting

The Hardy–Orlicz space \[H^{log}(\mathbb{D}) = \left\{ f: \mathbb{D} \rightarrow \mathbb{C} \text{ analytic}, \sup_{0\le r <1} \int_0^{2 \pi} \frac{|f(r e^{it}) }{ \log(e+ |f(re^{it})| )} dt < \infty\right \}\] was identified as the space of products \(BMOA(\mathbb{D}) \cdot H^1(\mathbb{D})\) in the work of Bonami and collaborators [3], creating a profound connection to the theory of Hankel operators. In this talk, we want to present a dyadic approach to \(H^{log}\) and to more general Hardy–Orlicz spaces, which was first considered in [1]. More recently, it has turned out that this approach allows a precise description of the enveloping Banach space, which in turn opens a path to the consideration of the product setting. This has so far been out of reach.

The talk is based on joint work with Odysseas Bakas (University of Patras, Greece), Salvador Rodríguez-López, and Alan Sola (both Stockholm University, Sweden).

• [1] O. Bakas et al., Notes on \(H^{\log}\): structural properties, dyadic variants, and bilinear \(H^1\)-\(BMO\) mappings, Ark. Mat. 60 (2022), no. 2, 231–275; MR4500365
• [2] O. Bakas et al., Multipliers for Hardy-Orlicz spaces and applications, J. Anal. Math. 155 (2025), no. 2, 401–458; MR4905316
• [3] A. Bonami et al., On the product of functions in BMO and \(H^1\), Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1405–1439; MR2364134
• [4] S. Pott and B. F. Sehba, Logarithmic mean oscillation on the polydisc, endpoint results for multi-parameter paraproducts, and commutators on BMO, J. Anal. Math. 117 (2012), 1–27; MR2944088