BMO functions and Carleson measures with values in uniformly convex spaces

TL;DR

This paper establishes a characterization of Banach spaces with uniformly convex norms via inequalities involving vector-valued BMO functions and Carleson measures, linking geometric properties of the space to harmonic analysis.

Contribution

It proves a new equivalence between the geometric property of uniform convexity in Banach spaces and certain inequalities involving vector-valued BMO functions and Carleson measures.

Findings

Characterizes uniformly convex Banach spaces through inequalities involving BMO functions.

Links geometric properties of Banach spaces to harmonic analysis tools.

Provides conditions under which inequalities involving gradients and Carleson measures hold.

Abstract

This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbb T$, respectively. For $1< q<\infty$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\T}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.