Martingale transform and Square function: some weak and restricted weak sharp weighted estimates

TL;DR

This paper investigates sharp weighted estimates for martingale transforms and square functions, demonstrating the existence of weights with large Muckenhoupt constants where these operators exhibit specific weak and restricted weak type bounds.

Contribution

It constructs explicit examples of weights in A_1 and A_2 classes where weak and restricted weak bounds for martingale transforms and square functions are sharp or nearly sharp, advancing understanding of weighted inequalities.

Findings

Existence of weights in A_1 with unbounded constants where martingale transforms have large weak (1,1) bounds.

Construction of weights in A_2 with large constants where square functions exhibit near-sharp bounds.

Quantitative relationships between A_2 constants, maximal functions, and square function bounds.

Abstract

Following the ideas of A. Lerner, F. Nazarov, S. Ombrosi from [12] we prove that there is a sequence of weights $w\in A^d_1$ such that $[w]^d_{A_1}\to \infty$, and martingale transforms $T$ such that with an absolute positive $c$ $\|T: L^1(w) \to L^{1, \infty}(w)\| \ge c [w]^d_{A_1}\log [w]^d_{A_1}$. We also show the existence of the sequence of weights (now in $A_2$) such that $[w]^d_{A_2}\to \infty$, and such that the following holds: $[w]_{A_2^d}\asymp \|M^d\|_{w^{-1}}^2$; $\|S_{w}: L^{2} (w) \to L^2(w^{-1})\| \ge c\, \|M^d\|_{w^{-1}}\sqrt{\log \|M^d\|_{w^{-1}}}$; $\|S_{w}: L^{2,1} (w) \to L^2(w^{-1})\| \ge c\, \|M^d\|_{w^{-1}}\sqrt{\log \|M^d\|_{w^{-1}}}; $ $\|S: L^2(w)\to L^{2, \infty}(w)\|=\|S_{w^{-1}}: L^{2}(w^{-1}) \to L^{2,\infty}(w)\|\le C\, \|M^d\|_{w^{-1}}\le C\, ([w]^d_{A_2})^{1/2}$. Finally, it is shown that for test functions of the form $\chi_I$ the weak norm $\|S_w \chi_I\|_{L^{2, \infty}} \le C\, [w]_{A_2^d}^{1/2}\|\chi_I\|_w$.