{"title":"Revised logarithmic Sobolev inequalities of fractional order","authors":"Marianna Chatzakou , Michael Ruzhansky","doi":"10.1016/j.bulsci.2024.103530","DOIUrl":null,"url":null,"abstract":"<div><div>In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with an explicit expression for the constant. Namely, we show that for all <span><math><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> and <span><math><mi>a</mi><mo>></mo><mn>0</mn></math></span> we have the inequality<span><span><span><math><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><mo>|</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mo>|</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mi>x</mi><mo>+</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo>(</mo><mn>1</mn><mo>+</mo><mi>log</mi><mo></mo><mi>a</mi><mo>)</mo><msubsup><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≤</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo>)</mo><msubsup><mrow><mo>‖</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msubsup></math></span></span></span> with an explicit <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo>)</mo></math></span> depending on <em>a</em>, the order <em>s</em>, and the dimension <em>n</em>, and investigate the behaviour of <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo>)</mo></math></span> for large <em>n</em>. Notably, for large <em>n</em> and when <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span>, the constant <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>a</mi><mo>)</mo></math></span> is asymptotically the same as the sharp constant of Lieb and Loss that was computed in <span><span>[13]</span></span>. Moreover, we prove a similar type inequality for functions <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>∩</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> whenever <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>n</mi></math></span> and <span><math><mi>p</mi><mo><</mo><mi>q</mi><mo>≤</mo><mfrac><mrow><mi>p</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>p</mi></mrow></mfrac></math></span>.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin des Sciences Mathematiques","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0007449724001489","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on with an explicit expression for the constant. Namely, we show that for all and we have the inequality with an explicit depending on a, the order s, and the dimension n, and investigate the behaviour of for large n. Notably, for large n and when , the constant is asymptotically the same as the sharp constant of Lieb and Loss that was computed in [13]. Moreover, we prove a similar type inequality for functions whenever and .