Failure of necessity of the energy condition
Classical Analysis and ODEs
2017-10-12 v3
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.
Keywords
Cite
@article{arxiv.1607.06071,
title = {Failure of necessity of the energy condition},
author = {Eric T. Sawyer and Chun-Yen Shen and Ignacio Uriarte-Tuero},
journal= {arXiv preprint arXiv:1607.06071},
year = {2017}
}
Comments
33 pages, typos corrected and some alterations suggested by referees, in particular correcting the definition of gradient elliptic