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arXiv · 0911.0713

Weak and Strong-type estimates for Haar Shift Operators: Sharp power on the $A_p$ characteristic

Tuomas P. Hytönen·Michael T. Lacey·Maria Carmen Reguera·Armen Vagharshakyan
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Abstract

As a corollary to our main result we deduce sharp A_p$ inequalities for T being either the Hilbert transform in dimension d=1, the Beurling transform in dimension d=2, or a Riesz transform in any dimension d\ge 2. For T_{\ast} the maximal truncations of these operators, we prove the sharp A_p weighted weak and strong-type L ^{p} (w) inequalities, for all 1<p<\infty. Key elements of the proof are (1) extrapolation (2) a recent argument for the A_2 bound in the untruncated case, an argument of Lacey-Petermichl-Reguera. (3) a weak-L^1 estimate for duals of maximal truncations. And (4) recent characterizations of the two-weight inequalities for strong and weak type inequalities, due to Lacey-Sawyer-Uriate-Tuero.

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Tuomas P. Hytönen, Michael T. Lacey, Maria Carmen Reguera, Armen Vagharshakyan. 2009-11-03. Weak and Strong-type estimates for Haar Shift Operators: Sharp power on the $A_p$ characteristic. https://arxiv.org/abs/0911.0713

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