Weak and Strong-type estimates for Haar Shift Operators: Sharp power on the $A_p$ characteristic
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.
Cite
@article{arxiv.0911.0713,
title = {Weak and Strong-type estimates for Haar Shift Operators: Sharp power on the $A_p$ characteristic},
author = {Tuomas P. Hytönen and Michael T. Lacey and Maria Carmen Reguera and Armen Vagharshakyan},
journal= {arXiv preprint arXiv:0911.0713},
year = {2011}
}
Comments
The paper has been withdrawn by the authors. The main results of this paper are improved and extended in arXiv:1103.5229. Also, Theorem 3.5 of 0911.0713 is correct, but the proof requires elaboration