Sawyer E.TCHUN-YEN SHENUriarte-Tuero I.2022-11-112022-11-11202110737928https://www.scopus.com/inward/record.uri?eid=2-s2.0-85102711710&doi=10.1093%2fimrn%2frnz269&partnerID=40&md5=58aeb0b6bd1a67b533d88f30348ed69dhttps://scholars.lib.ntu.edu.tw/handle/123456789/624584We show that the energy conditions are not necessary for boundedness of Riesz transforms in dimension $n\geq 2$. In dimension $n=1$, we construct an elliptic singular integral operator $H_{\flat } $ for which the energy conditions are not necessary for boundedness of $H_{\flat }$. The convolution kernel $K_{\flat }\left (x\right) $ of the operator $H_{\flat }$ is a smooth flattened version of the Hilbert transform kernel $K\left (x\right) =\frac{1}{x}$ that satisfies ellipticity $ \vert K_{\flat }\left (x\right) \vert \gtrsim \frac{1}{\left \vert x\right \vert }$, but not gradient ellipticity $ \vert K_{\flat }^{\prime }\left (x\right) \vert \gtrsim \frac{1}{ \vert x \vert ^{2}}$. Indeed the kernel has flat spots where $K_{\flat }^{\prime }\left (x\right) =0$ on a family of intervals, but $K_{\flat }^{\prime }\left (x\right) $ is otherwise negative on $\mathbb{R}\setminus \left \{ 0\right \} $. On the other hand, if a one-dimensional kernel $K\left (x,y\right) $ is both elliptic and gradient elliptic, then the energy conditions are necessary, and so by our theorem in [30], the $T1$ theorem holds for such kernels on the line. This paper includes results from arXiv:16079.06071v3 and arXiv:1801.03706v2. © 2019 The Author(s) 2019. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.[SDGs]SDG7journal article10.1093/imrn/rnz2692-s2.0-85102711710