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arXiv · 1308.5349

The Linear Bound for the Natural Weighted Resolution of the Haar Shift

Sandra Pott·Maria Carmen Reguera·Eric T. Sawyer·Brett D. Wick
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Abstract

The Hilbert transform has a linear bound in the $A_{2}$ characteristic on weighted $L^{2}$, \begin{equation*} \left\Vert H\right\Vert _{L^{2}(w)\rightarrow L^{2}(w)}\lesssim \left[ w \right] _{A_{2}}, \end{equation*} and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation $M_{w^{\frac{1}{2}}}\mathcal{S }M_{w^{-\frac{1}{2}}}$ induced by the canonical decomposition of a multiplier into paraproducts:% \begin{equation*} M_{f}=P_{f}^{-}+P_{f}^{0}+P_{f}^{+}. \end{equation*} The main tools used are composition of paraproducts, a product formula for Haar coefficients, the Carleson Embedding Theorem, and the linear bound for the square function.

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Sandra Pott, Maria Carmen Reguera, Eric T. Sawyer, Brett D. Wick. 2014-01-12. The Linear Bound for the Natural Weighted Resolution of the Haar Shift. https://arxiv.org/abs/1308.5349

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