A two weight fractional singular integral theorem with side conditions, energy and k-energy dispersed
Journal
Association for Women in Mathematics Series
Journal Volume
5
Pages
305-372
Date Issued
2017
Author(s)
Abstract
This paper is a sequel to our paper Sawyer et al. (Revista Mat Iberoam 32(1):79–174, 2016). Let σ and ω be locally finite positive Borel measures on Rn (possibly having common point masses), and let Tα be a standard α-fractional Calderón-Zygmund operator on Rn with 0 ≤ α < n. Suppose that Ω : Rn→ ℝn is a globally biLipschitz map, and refer to the images Ω Q of cubes Q as quasicubes. Furthermore, assume as side conditions the 𝒜2α conditions, punctured A2α conditions, and certain α -energy conditions taken over quasicubes. Then we show that Tα is bounded from L2(σ) to L2(ω) if the quasicube testing conditions hold for Tα and its dual, and if the quasiweak boundedness property holds for Tα. Conversely, if Tα is bounded from L2(σ) to L2(ω), then the quasitesting conditions hold, and the quasiweak boundedness condition holds. If the vector of α-fractional Riesz transforms Rσα (or more generally a strongly elliptic vector of transforms) is bounded from L2(σ) to L2(ω), then both the 𝒜2α conditions and the punctured A2α conditions hold. Our quasienergy conditions are not in general necessary for elliptic operators, but are known to hold for certain situations in which one of the measures is one-dimensional (Lacey et al., Two weight inequalities for the Cauchy transform from R to C+, arXiv:1310.4820v4; Sawyer et al., The two weight T1 theorem for fractional Riesz transforms when one measure is supported on a curve, 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 (Two weight inequalities for the Cauchy transform from R to C+, arXiv:1310.4820v1, versions 2 and 3). © The Author(s) and the Association for Women in Mathematics 2017.
SDGs
Type
book part