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arXiv · 1402.5523

The Linear Bound for Haar Multiplier Paraproducts

Kelly Bickel·Eric T. Sawyer·Brett D. Wick
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Abstract

We study the natural resolution of the conjugated Haar multiplier $M_{w^{\frac{1}{2}}}T_{\sigma}M_{w^{-\frac{1}{2}}},$ where the multiplication operators $M_{w^{\pm\frac{1}{2}}}$ are decomposed into their canonical paraproduct decompositions. We prove that each constituent operator obtained from this resolution has a linear bound on $L^2(\mathbb{R}^d;w)$ in terms of the $A_{2}$ characteristic of $w$. The main tools used are a product formula for Haar coefficients, the Carleson Embedding Theorem, the linear bound for the square function, and the well-known linear bound of $T_{\sigma}$ on $L^2(\mathbb{R}^d,w).$

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Kelly Bickel, Eric T. Sawyer, Brett D. Wick. 2014-02-22. The Linear Bound for Haar Multiplier Paraproducts. https://arxiv.org/abs/1402.5523

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