{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:41:21Z","timestamp":1760197281517,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,3,23]],"date-time":"2018-03-23T00:00:00Z","timestamp":1521763200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate the exponential convergence of a Markovian semigroup in the Zygmund space under the assumption of logarithmic Sobolev inequality. We show that the convergence rate is greater than the logarithmic Sobolev constant. To do this, we use the notion of entropy. We also give an example of a Laguerre operator. We determine the spectrum in the Orlicz space and discuss the relation between the logarithmic Sobolev constant and the spectral gap.<\/jats:p>","DOI":"10.3390\/e20040220","type":"journal-article","created":{"date-parts":[[2018,3,27]],"date-time":"2018-03-27T23:37:17Z","timestamp":1522193837000},"page":"220","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Logarithmic Sobolev Inequality and Exponential Convergence of a Markovian Semigroup in the Zygmund Space"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8865-8071","authenticated-orcid":false,"given":"Ichiro","family":"Shigekawa","sequence":"first","affiliation":[{"name":"Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bakry, D., Gentil, I., and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators, Springer. Grundlehren der Mathematischen Wissenschaften 348.","DOI":"10.1007\/978-3-319-00227-9"},{"key":"ref_2","unstructured":"Deuschel, J.-D., and Stroock, D.W. (1989). Large Deviations, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.jfa.2006.10.007","article-title":"Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups","volume":"243","author":"Roberto","year":"2007","journal-title":"J. Funct. Anal."},{"key":"ref_4","unstructured":"Rao, M.M., and Ren, Z.D. (1991). Theory of Orlicz Spaces, Marcel Dekker. Monographs and Textbooks in Pure and Applied Mathematics, 146."},{"key":"ref_5","unstructured":"Engel, K.-J., and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations, Springer."},{"key":"ref_6","unstructured":"Lebedev, N.N. (1972). Special Functions and Their Applications, Dover."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1090\/S0002-9939-1985-0781062-0","article-title":"An example in the theory of hypercontractive semigroups","volume":"94","author":"Korzeniowski","year":"1985","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Beals, R., and Wong, R. (2010). Special Functions, Cambridge University Press.","DOI":"10.1017\/CBO9780511762543"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1215\/21562261-2642431","article-title":"Exponential convergence of Markovian semigroups and their spectra on Lp-spaces","volume":"54","author":"Kusuoka","year":"2014","journal-title":"Kyoto J. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/220\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:58:18Z","timestamp":1760194698000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/220"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,23]]},"references-count":9,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["e20040220"],"URL":"https:\/\/doi.org\/10.3390\/e20040220","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,3,23]]}}}