Directional Poincaré inequalities along mixing flows

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• 作者：Stefan Steinerberger
• 刊名：Arkiv f?r Matematik
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：54
• 期：2
• 页码：555-569
• 全文大小：977 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1871-2487
• 卷排序：54

文摘

We provide a refinement of the Poincaré inequality on the torus \(\mathbb{T}^{d}\): there exists a set \(\mathcal{B} \subset \mathbb{T} ^{d}\) of directions such that for every \(\alpha \in \mathcal{B}\) there is a \(c_{\alpha } > 0\) with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative \(\langle \nabla f, \alpha \rangle \) does not detect any oscillation in directions orthogonal to \(\alpha \), however, for certain \(\alpha \) the geodesic flow in direction \(\alpha \) is sufficiently mixing to compensate for that defect. On the two-dimensional torus \(\mathbb{T}^{2}\) the inequality holds for \(\alpha = (1, \sqrt{2})\) but is not true for \(\alpha = (1,e)\). Similar results should hold at a great level of generality on very general domains.References1. Cassels, J. W. S., An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge University Press, Cambridge, 1957.MATHGoogle Scholar2. Cusick, T. W. and Flahive, M. E., The Markov and Lagrange Spectra, Am. Math. Soc., Providence, RI, 1989.CrossRefMATHGoogle Scholar3. Dickinson, H., The Hausdorff dimension of systems of simultaneously small linear forms, Mathematika 40 (1993), 367–374.MathSciNetCrossRefMATHGoogle Scholar4. Hurwitz, A., Ueber die angenaeherte Darstellung der Irrationalzahlen durch rationale Brueche, Math. Ann. 39 (1891), 279–284.MathSciNetCrossRefGoogle Scholar5. Hussain, M., A note on badly approximable linear forms, Bull. Aust. Math. Soc. 83 (2011), 262–266.MathSciNetMATHGoogle Scholar6. Hussain, M. and Kristensen, S., Badly approximable systems of linear forms in absolute value, Unif. Distrib. Theory 8 (2013), 7–15.MathSciNetMATHGoogle Scholar7. Iosifescu, M. and Kraaikamp, C., Metrical Theory of Continued Fractions, Mathematics and Its Applications 547, Kluwer Academic, Dordrecht, 2002.CrossRefMATHGoogle Scholar8. Khinchine, A., Continued Fractions, University of Chicago Press, Chicago, Il, 1964.Google Scholar9. Khintchine, A., Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), 706–714.MathSciNetCrossRefMATHGoogle Scholar10. Marcovecchio, R., The Rhin–Viola method for \(\log {2}\), Acta Arith. 139 (2009), 147–184.MathSciNetCrossRefMATHGoogle Scholar11. Perron, O., Über diophantische Approximationen, Math. Ann. 84 (1921), 77–84.CrossRefMATHGoogle Scholar12. Salikhov, V. Kh., On the irrationality measure of \(\pi \), Uspekhi Mat. Nauk 63 (2008), 163–164. translation in Russian Math. Surveys 63 (2008), 570–572.MathSciNetCrossRefGoogle Scholar13. Schmidt, W., Badly approximable systems of linear forms, J. Number Theory 1 (1969), 139–154.MathSciNetCrossRefMATHGoogle ScholarCopyright information© Institut Mittag-Leffler 2016Authors and AffiliationsStefan Steinerberger1Email author1.Department of MathematicsYale UniversityNew HavenU.S.A. About this article CrossMark Print ISSN 0004-2080 Online ISSN 1871-2487 Publisher Name Springer Netherlands About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s11512-016-0241-7_Directional Poincaré inequalities ", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s11512-016-0241-7_Directional Poincaré inequalities ", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips