On $A_p$-$A_\infty$ type estimates for square functions

TL;DR

This paper establishes improved strong-type $A_p$-$A_ Infty$ estimates for square functions, utilizing advanced techniques like entropy bounds, stopping cubes, and pigeon-hole arguments to enhance previous bounds.

Contribution

It introduces new $A_p$-$A_ Infty$ estimates for square functions and applies recent entropy bound methods, advancing the theoretical understanding of these operators.

Findings

Proved strong-type $A_p$-$A_ Infty$ estimate for square functions.

Established entropy bounds using Treil-Volberg's approach.

Enhanced previous bounds by Lerner.

Abstract

We prove strong-type $A_p$-$A_\infty$ estimate for square functions, improving on the $ A_p$ bound due to Lerner. Entropy bounds, in the recent innovation of Treil-Volberg, are then proved. The techniques of proof include parallel stopping cubes, pigeon-hole arguments, and the approach to entropy bounds of Lacey--Spencer.