Square function characterization of Hardy-type spaces

2026/07/24 by Evgeny Abakumov, Evgueni Doubtsov, Dmitry Rutsky
#math.FA #math.CV

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Abstract

The well-known characterization of Hardy spaces H_p(\mathbb D), 0 < p < ∞, in terms of the Littlewood-Paley g-function S f (ζ) = (∫₀¹ |f' (r ζ)|² (1 - r) \mathrm dr)^1/2 ∈ L_p is generalized to Hardy-type spaces X_A corresponding to quasi-Banach lattices X on the unit circle \mathbb T under the assumption that the Hardy-Littlewood maximal operator M is bounded in (X^δ)' with some δ> 0. As an application to composition operators C_φ, we derive an exact criterion for the boundedness and compactness of C_φ: B^ω→ X_A, where B^ω= \f | sup |f'|/ω< ∞\ is the weighted Bloch space with a log-convex radial weight ω, generalizing recent results in the one-dimensional setting.

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