Uniformly analytic convex spaces

Abstract

Let X be a complex quasi-Banach space and 2 ≤ q < ∞. We prove that X is analytically q-uniformly convexifiable (i.e. admits an equivalent uniformly analytic convex quasi-norm of power type q) if and only if, for every symmetric quasi-Banach function space E on (0, ∞) with the Fatou property whose dilation operators satisfy ‖Da‖ → 0 as a → 0+, the q-th power of the q-square function S(q)(F) (respectively, of the conditional q-square function σ(q)(F)) of every X-valued Hardy martingale F is dominated in the norm of E by the q-th power of the corresponding sharp maximal function. The proof combines a stopping-time good-λ estimate for Hardy martingales with the transfer lemma of Alday and Dauitbek, which converts good-λ inequalities into norm inequalities in symmetric spaces. The result extends the known characterization in Orlicz spaces with a moderate Young function and yields new characterizations in Lebesgue spaces Lr (0 < r < ∞) and Lorentz spaces.

Keywords: uniformly analytic convex space, analytic q-uniform convexity, Hardy martingale, q-square function, sharp maximal function, symmetric quasi-Banach function space, good-λ inequality, Orlicz space, Lorentz space

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Как цитировать

Anuar Uten Uniformly analytic convex spaces // Central Asian Scientific Journal. – 2026. – No. 4(32), Vol. 2. – Pp. 385–391.