Best constants for a family of Carleson sequences

TL;DR

This paper determines sharp bounds for Carleson sequences related to dyadic A_2 weights, generalizing previous estimates and providing explicit optimizers through Bellman function techniques.

Contribution

It introduces precise bounds for Carleson norms in terms of A_2-characteristics, extending earlier work and offering explicit optimizer constructions.

Findings

Sharp bounds for Carleson sequences in terms of A_2 weights

Explicit Bellman function expressions and bounds

New characterizations of dyadic A_2 weights

Abstract

We consider a general family of Carleson sequences associated with dyadic $A_2$ weights and find sharp -- or, in one case, simply best known -- upper and lower bounds for their Carleson norms in terms of the $A_2$-characteristic of the weight. The results obtained make precise and significantly generalize earlier estimates by Wittwer, Vasyunin, Beznosova, and others. We also record several corollaries, one of which is a range of new characterizations of dyadic $A_2.$ Particular emphasis is placed on the relationship between sharp constants and optimizing sequences of weights; in most cases explicit optimizers are constructed. Our main estimates arise as consequences of the exact expressions, or explicit bounds, for the Bellman functions for the problem, and the paper contains a measure of Bellman-function innovation.