Failure of necessity of the energy condition

Abstract

We give an example of a pair of weights (u,v) on the line, and an elliptic convolution singular integral operator H on the line, such that H_u is bounded from L^2(u) to L^2(v), yet the measure pair (u,v) fails to satisfy the backward energy condition. The key to the construction is that the kernel K of H has flat spots where d/dx K(x) = 0. Conversely, we show that if H is gradient elliptic, i.e. d/dx K(x) =< c < 0, then the energy conditions are necessary for boundedness of H, and by our theorem in arXiv:1603.04332v2, the T1 theorem holds for H.

33 pages, typos corrected and some alterations suggested by referees, in particular correcting the definition of gradient elliptic

References in corpus (4)

• 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 with an energy side condition, quasicube testing and common point masses

Cited by in corpus (1)

• Necessary/sufficient conditions in weighted theory