A two weight fractional singular integral theorem with side conditions, energy and k-energy dispersed

TL;DR

This paper establishes a two-weight boundedness criterion for fractional Calderón-Zygmund operators under side conditions, including energy and k-energy dispersion, extending previous results and exploring necessary conditions in higher dimensions.

Contribution

It introduces new side conditions like k-energy dispersion and provides a comprehensive two-weight boundedness theorem for fractional singular integrals.

Findings

Boundedness characterized by quasitesting and quasiweak boundedness conditions.

Energy and k-energy dispersion conditions are sufficient for boundedness.

Results extend to vector-valued fractional Riesz transforms under certain conditions.

Abstract

This paper is a sequel to our paper Rev. Mat. Iberoam. 32 (2016), no. 1, 79-174. Let T be a standard fractional Calderon Zygmund operator. Assume appropriate Muckenhoupt and quasienergy side conditions. Then we show that T is bounded from one weighted space to another if the quasicube testing conditions hold for T and its dual, and if the quasiweak boundedness property holds for T. Conversely, if T is bounded, then the quasitesting conditions hold, and the quasiweak boundedness condition holds. If the vector of fractional Riesz transforms (or more generally a strongly elliptic vector of transforms) is bounded, then the appropriate Muckenhoupt conditions hold. We do not know if our quasienergy conditions are necessary in higher dimensions, except for certain situations in which one of the measures is one-dimensional as in arXiv:1310.4820 and arXiv:1505.07822v4, and for certain side conditions placed on the measures such as doubling and k-energy dispersed, which when k=n-1 is similar to the condition of uniformly full dimension in Lacey and Wick arXiv:1312.6163v3.