On the failure of lower square function estimates in the non-homogeneous weighted setting

TL;DR

This paper investigates the limitations of classical weighted conditions for lower square function estimates in non-homogeneous settings, revealing new bounds and contrasts with homogeneous cases.

Contribution

It demonstrates the insufficiency of the $A_{ infty}$ condition and establishes new bounds under the martingale $A_2$ condition, highlighting differences from homogeneous cases.

Findings

Classical $A_{ infty}$ condition is not enough for lower square function estimates.

Under martingale $A_2$, estimates hold but with a higher characteristic power.

Sharp $A_{ infty}$ estimate for $n$-adic homogeneous case grows with $n$.

Abstract

We show that the classical $A_{\infty}$ condition is not sufficient for a lower square function estimate in the non-homogeneous weighted $L^2$ space. We also show that under the martingale $A_2$ condition, an estimate holds true, but the optimal power of the characteristic jumps from $1 / 2$ to $1$ even when considering the classical $A_2$ characteristic. This is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete timenon-homogeneous martingale transforms. Last, we give a sharp $A_{\infty}$ estimate for the $n$-adic homogeneous case, growing with $n$.