{"paper":{"title":"Failure of necessity of the energy condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Chun-Yen Shen, Eric T. Sawyer, Ignacio Uriarte-Tuero","submitted_at":"2016-07-20T19:37:44Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06071","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}