{"paper":{"title":"Centered Sobolev inequality and exponential convergence in $\\Phi$-entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Liming Wu, Lingyan Cheng","submitted_at":"2017-03-01T20:52:41Z","abstract_excerpt":"In this short paper we find that the Sobolev inequality $$\\frac 1{p-2}\\left[\\left(\\int f^{p} d\\mu\\right)^{2/p} - \\int f^2 d\\mu\\right] \\le C \\int |\\nabla f|^2 d\\mu$$ ($p\\ge 0$) is equivalent to the exponential convergence of the Markov diffusion semigroup $(P_t)$ to the invariant measure $\\mu$, in some $\\Phi$-entropy. We provide the estimate of the exponential convergence in total variation and a bounded perturbation result under the Sobolev inequality. Finally in the one-dimensional case we get some two-sided estimates of the Sobolev constant by means of the generalized Hardy inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00491","kind":"arxiv","version":1},"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"}