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arXiv · 1103.5229

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

Tuomas P. Hytönen·Michael T. Lacey·Henri Martikainen·Tuomas Orponen·Maria Carmen Reguera·Eric T. Sawyer·Ignacio Uriarte-Tuero
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Abstract

For 1<p< \infty, weight w \in A_p, and any L ^2 -bounded Calderón-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 Hytönen-Pérez-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.

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Tuomas P. Hytönen, Michael T. Lacey, Henri Martikainen, Tuomas Orponen, Maria Carmen Reguera, Eric T. Sawyer, Ignacio Uriarte-Tuero. 2011-03-27. Weak And Strong Type Estimates for Maximal Truncations of Calderón-Zygmund Operators on $ A_p$ Weighted Spaces. https://arxiv.org/abs/1103.5229

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