A restricted weak type inequality with application to a Tp theorem and optimal cancellation conditions for CZO's
arXiv:1907.07571
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.
7 pages. Typos corrected, especially changing classical A2 to one-tailed A2 in some statements. Results otherwise unchanged. Continues the investigation begun in arXiv:1906.05602