Weighted square function estimates

TL;DR

This paper proves explicit, optimal weighted norm inequalities for martingale and classical square functions in harmonic analysis, using the Bellman function method to achieve bounds that depend precisely on weight characteristics and p.

Contribution

It introduces new explicit bounds for weighted square function inequalities that are optimal in both weight dependence and the limit as p approaches infinity.

Findings

Bounds are explicit and optimal for weight characteristics.

Results hold for both martingale and classical square functions.

Bounds are sharp as p approaches infinity.

Abstract

The paper contains the proof of $L^p$-weighted norm inequalities for both, martingales square functions and the classical square functions in harmonic analysis of Littlewood-Paley and Lusin. Furthermore, the bounds are completely explicit and are optimal not only on the dependence of the characteristics of the weight but also on the dependance on $p$, as $p\to\infty$. The proof rests on Bellman function method: the estimates are deduced from the existence of an appropriate and rather complicated function of four variables.