Weak And Strong Type Estimates for Maximal Truncations of Calder\'on-Zygmund Operators on $ A_p$ Weighted Spaces

TL;DR

This paper establishes sharp weighted norm inequalities for maximal truncations of Calderón-Zygmund operators on A_p weighted spaces, demonstrating optimal dependence on the A_p characteristic without using extrapolation.

Contribution

It provides the first sharp bounds for both weak and strong type norms of maximal truncations of Calderón-Zygmund operators on A_p weights, with new techniques including weak-type estimates and two-weight conditions.

Findings

Weak-type norm of T_# is controlled by  w_{A_p}.

Strong-type norm depends on  w_{A_p} to the power  max(1, (p-1)^{-1}).

Estimates are proven to be sharp and cannot be improved in the power of  w_{A_p}.

Abstract

For 1<p< \infty, weight w \in A_p, and any L ^2 -bounded Calder\'on-Zygmund operator T, we show that there is a constant C(T,P) so that we prove the sharp norm dependence on T_#, the maximal truncations of T, in both weak and strong type L^p(w) norms. Namely, for the weak type norm, T_# maps L^p(w) to weak-L^p(w) with norm at most \|w\|_{A_p}. And for the strong type norm, the norm estimate is \|w\|_{A_p}^{\max(1, (p-1) ^{-1})}. These estimates are not improvable in the power of \lVert w\rVert_{A_p}. Our argument follows the outlines of the arguments of Lacey-Petermichl-Reguera (Math.\ Ann.\ 2010) and Hyt\"onen-P\'erez-Treil-Volberg (arXiv, 2010) with new ingredients, including a weak-type estimate for certain duals of T_#, and sufficient conditions for two weight inequalities in L ^{p} for T_#. Our proof does not rely upon extrapolation.