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arXiv · 1907.07571

A restricted weak type inequality with application to a Tp theorem and optimal cancellation conditions for CZO's

Eric T. Sawyer
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Abstract

This paper continues the investigation begun in arXiv:1906.05602 of extending the T1 theorem of David and Journé, and optimal cancellation conditions, to more general weight pairs. The main additional tool developed here is a two weight restricted weak type inequality. Assuming σ and ω are locally finite positive Borel measures, with one of them an A infinity weight, we show that the two weight restricted weak type inequality for an α-fractional Calderón-Zygmund singular integral T holds if and (provided T is elliptic) only if the classical fractional Muckenhoupt condition holds. In the case α = 0, boundedness of T on unweighted L2 is assumed as well. Applications are then given to a Tp theorem for CZO's with doubling weights when one is A infinity, and then to optimal cancellation conditions for CZO's in terms of polynomial testing in similar situations.

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Eric T. Sawyer. 2019-10-23. A restricted weak type inequality with application to a Tp theorem and optimal cancellation conditions for CZO's. https://arxiv.org/abs/1907.07571

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