{"paper":{"title":"Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\\mathbb{R}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Can Zhang, Gengsheng Wang, Ming Wang, Yubiao Zhang","submitted_at":"2017-11-12T11:58:06Z","abstract_excerpt":"This paper studies connections among observable sets, the observability inequality, the H\\\"{o}lder-type interpolation inequality and the spectral inequality for the heat equation in $\\mathbb R^n$. We present a characteristic of observable sets for the heat equation. In more detail, we show that a measurable set in $\\mathbb{R}^n$ satisfies the observability inequality if and only if it is $\\gamma$-thick at scale $L$ for some $\\gamma>0$ and $L>0$.We also build up the equivalence among the above-mentioned three inequalities. More precisely, we obtain that if a measurable set $E\\subset\\mathbb{R}^n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04279","kind":"arxiv","version":2},"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"}