A geometric condition, necessity of energy, and two weight boundedness of fractional Riesz transforms

Abstract

We prove that the energy conditions in arXiv:1302.5093v7 are implied by the fractional A2 conditions and testing conditions for the vector of fractional Riesz transforms when one measure is supported on a line. Then we apply the main theorem in arXiv:1302.5093v7 to obtain a T1 theorem for the fractional Riesz transforms when one weight is supported on a line.

21 pages. We thank Michael Lacey for pointing out our failure to address the weak boundedness property in an earlier version. This version uses the corrected theorem in arXiv:1302.5093v7, which in turn uses the Lacey-Wick Monotonicity Lemma

References in corpus (5)

• Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures
• Two Weight Inequalities for Riesz Transforms: Uniformly Full Dimension Weights
• Two Weight Inequalities for the Cauchy Transform from $\mathbb{R}$ to ${\mathbb{C}}_{+}$
• A two weight theorem for fractional singular integrals in higher dimension
• A note on failure of energy reversal for classical fractional singular integrals

Cited by in corpus (3)

• Energy conditions and twisted localizations of operators
• A note on failure of energy reversal for classical fractional singular integrals
• Characterization of a class of embeddings for function spaces with Muckenhoupt weights