2019/06/13 by Eric T. Sawyer, Sawyer, Eric T. · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.CA
paper · pdf · doi:10.48550/arxiv.1906.05602
openalex publication_date 2019/06/13 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We begin an investigation into extending the T1 theorem of David and Journé, and the corresponding cancellation conditions of Stein, to more general pairs of distinct doubling weights. For example, assuming the measures satisfy a fractional A infinity condition and are comparable in the sense of Coifman and Fefferman, we characterize the two weight norm inequality for a strongly elliptic fractional Calderón-Zygmund singular integral, in terms of the one-tailed fractional Muckenhoupt conditions, and the usual cube testing conditions. We then apply this result to give a version, in the setting of two comparable fractional A infinity weights, of Stein's characterization of cancellation conditions on a kernel K in order that there exists a bounded operator T that is associated with K. More generally we prove a T1 theorem involving a bilinear indicator/cube testing inequality in place of the weak boundedness property of David and Journeé - where we must test over all bounded functions instead of just Holder continuous functions. We use a proof strategy based on an adaptation of the `pivotal' argument of Nazarov, Treil and Volberg to the weighted Alpert wavelets of Rahm, Sawyer and Wick using a Parallel Corona decomposition of Lacey, Sawyer, Shen and Uriarte-Tuero.