Sharp Estimates for Lipschitz Class

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• 作者：Adam Osȩkowski
• 关键词：Sharp function ; Lipschitz ; Best constants
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：26
• 期：2
• 页码：1346-1369
• 全文大小：543 KB
• 参考文献：1.Burkholder, D.L.: A nonlinear partial differential equation and the unconditional constant of the Haar system in \(L^p\) . Bull. Am. Math. Soc. 7, 591–595 (1982)MathSciNet CrossRef MATH
  2.Burkholder, D.L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12, 647–702 (1984)MathSciNet CrossRef MATH
  3.Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Am. Math. Soc. 77, 587–588 (1971)MathSciNet CrossRef MATH
  4.Ivanishvili, P., Osipov, N.N., Stolyarov, D.M., Vasyunin, V., Zatitskiy, P.B.: On Bellman function for extremal problems in BMO. C. R. Math. Acad. Sci. Paris 350(11–12), 561–564 (2012)MathSciNet CrossRef MATH
  5.John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)MathSciNet CrossRef MATH
  6.Kovač, V.: Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J. 60(3), 813–846 (2011)MathSciNet CrossRef MATH
  7.Nazarov, F.L., Treil, S.R.: The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. St. Petersburg Math. J. 8, 721–824 (1997)MathSciNet
  8.Nazarov, F.L., Treil, S.R., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Am. Math. Soc. 12, 909–928 (1999)MathSciNet CrossRef MATH
  9.Osȩkowski, A.: Sharp martingale and semimartingale inequalities. Monografie Matematyczne, vol. 72. Birkhäuser, Basel (2012)CrossRef MATH
  10.Osȩkowski, A.: Sharp inequalities for BMO functions. Chin. Ann. Math. Ser B 36, 225–236 (2015)MathSciNet CrossRef
  11.Pereyra, M.C.: Haar multipliers meet Bellman functions. Rev. Mat. Iberoam. 25(3), 799–840 (2009)MathSciNet CrossRef MATH
  12.Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical \(A_p\) characteristic. Am. J. Math. 129(5), 1355–1375 (2007)MathSciNet CrossRef MATH
  13.Petermichl, S., Wittwer, J.: A sharp estimate for the weighted Hilbert transform via Bellman functions. Michigan Math. J. 50(1), 71–87 (2002)MathSciNet CrossRef MATH
  14.Rey, G., Reznikov, A.: Extremizers and sharp weak-type estimates for positive dyadic shifts. Adv. Math. 254, 664–681 (2014)MathSciNet CrossRef MATH
  15.Slavin, L., Vasyunin, V.: Sharp results in the integral-form John-Nirenberg inequality. Trans. Am. Math. Soc. 363, 4135–4169 (2011)MathSciNet CrossRef MATH
  16.Slavin, L., Vasyunin, V.: Sharp \(L_p\) estimates on BMO. Indiana Univ. Math. J. 61(3), 1051–1110 (2012)MathSciNet CrossRef MATH
  17.Vasyunin, V.: The sharp constant in John-Nirenberg inequality, Preprint PDMI no. 20, 2003. http://​www.​pdmi.​ras.​ru/​preprint/​2003/​index.​html
  18.Vasyunin, V., Volberg, A.: The Bellman function for certain two weight inequality: the case study. St. Petersburg Math. J. 18, 201–222 (2007)MathSciNet CrossRef MATH
  19.Vasyunin, V., Volberg, A.: Monge-Ampère equation and Bellman optimization of Carleson Embedding Theorems, Am. Math. Soc. Transl. (2), vol. 226, “Linear and Complex Analysis”, pp. 195–238 (2009)
  20.Vasyunin, V., Volberg, A.: Sharp constants in the classical weak form of the John-Nirenberg inequality, preprint available at arxiv.​org/​pdf/​1204.​1782
  21.Wittwer, J.: A sharp estimate on the norm of the martingale transform. Math. Res. Lett. 7(1), 1–12 (2000)MathSciNet CrossRef MATH
  
• 作者单位：Adam Osȩkowski (1)
  
  1. Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

Let \(I\) be an interval contained in \({\mathbb {R}}\). For a given function \(f:I\rightarrow {\mathbb {R}}\), \(u\in I\) and any \(0<\alpha \le 1\), set $$\begin{aligned} f_\alpha ^\sharp (u)=\sup \frac{1}{|J|^{\alpha }}\left( \frac{1}{|J|}\int _J \left| f(x)-\frac{1}{|J|}\int _J f(y)\text{ d }y\right| ^2\text{ d }x\right) ^{1/2}, \end{aligned}$$where the supremum is taken over all subintervals \(J\subseteq I\) which contain \(u\). The paper contains the proofs of the estimates $$\begin{aligned} \ell (\alpha )\big |\big |f_\alpha ^\sharp \big |\big |_{L^\infty (I)}\le \big |\big |f\big |\big |_{{\text {Lip}}_\alpha (I)}\le L(\alpha )\big |\big |f_\alpha ^\sharp \big |\big |_{L^\infty (I)}, \end{aligned}$$where $$\begin{aligned} \ell (\alpha )=2\sqrt{2\alpha +1},\quad L(\alpha )=\frac{(4\alpha +4)^{(\alpha +1)/(2\alpha +1)}\sqrt{2\alpha +1}}{2\alpha } \end{aligned}$$are the best possible. The proof rests on the evaluation of Bellman functions associated with the above estimates. Keywords Sharp function Lipschitz Best constants Mathematics Subject Classification 26A16 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (21) References1.Burkholder, D.L.: A nonlinear partial differential equation and the unconditional constant of the Haar system in \(L^p\). Bull. Am. Math. Soc. 7, 591–595 (1982)MathSciNetCrossRefMATH2.Burkholder, D.L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12, 647–702 (1984)MathSciNetCrossRefMATH3.Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Am. Math. Soc. 77, 587–588 (1971)MathSciNetCrossRefMATH4.Ivanishvili, P., Osipov, N.N., Stolyarov, D.M., Vasyunin, V., Zatitskiy, P.B.: On Bellman function for extremal problems in BMO. C. R. Math. Acad. Sci. Paris 350(11–12), 561–564 (2012)MathSciNetCrossRefMATH5.John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)MathSciNetCrossRefMATH6.Kovač, V.: Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J. 60(3), 813–846 (2011)MathSciNetCrossRefMATH7.Nazarov, F.L., Treil, S.R.: The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. St. Petersburg Math. J. 8, 721–824 (1997)MathSciNet8.Nazarov, F.L., Treil, S.R., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Am. Math. Soc. 12, 909–928 (1999)MathSciNetCrossRefMATH9.Osȩkowski, A.: Sharp martingale and semimartingale inequalities. Monografie Matematyczne, vol. 72. Birkhäuser, Basel (2012)CrossRefMATH10.Osȩkowski, A.: Sharp inequalities for BMO functions. Chin. Ann. Math. Ser B 36, 225–236 (2015)MathSciNetCrossRef11.Pereyra, M.C.: Haar multipliers meet Bellman functions. Rev. Mat. Iberoam. 25(3), 799–840 (2009)MathSciNetCrossRefMATH12.Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical \(A_p\) characteristic. Am. J. Math. 129(5), 1355–1375 (2007)MathSciNetCrossRefMATH13.Petermichl, S., Wittwer, J.: A sharp estimate for the weighted Hilbert transform via Bellman functions. Michigan Math. J. 50(1), 71–87 (2002)MathSciNetCrossRefMATH14.Rey, G., Reznikov, A.: Extremizers and sharp weak-type estimates for positive dyadic shifts. Adv. Math. 254, 664–681 (2014)MathSciNetCrossRefMATH15.Slavin, L., Vasyunin, V.: Sharp results in the integral-form John-Nirenberg inequality. Trans. Am. Math. Soc. 363, 4135–4169 (2011)MathSciNetCrossRefMATH16.Slavin, L., Vasyunin, V.: Sharp \(L_p\) estimates on BMO. Indiana Univ. Math. J. 61(3), 1051–1110 (2012)MathSciNetCrossRefMATH17.Vasyunin, V.: The sharp constant in John-Nirenberg inequality, Preprint PDMI no. 20, 2003. http://​www.​pdmi.​ras.​ru/​preprint/​2003/​index.​html 18.Vasyunin, V., Volberg, A.: The Bellman function for certain two weight inequality: the case study. St. Petersburg Math. J. 18, 201–222 (2007)MathSciNetCrossRefMATH19.Vasyunin, V., Volberg, A.: Monge-Ampère equation and Bellman optimization of Carleson Embedding Theorems, Am. Math. Soc. Transl. (2), vol. 226, “Linear and Complex Analysis”, pp. 195–238 (2009)20.Vasyunin, V., Volberg, A.: Sharp constants in the classical weak form of the John-Nirenberg inequality, preprint available at arxiv.​org/​pdf/​1204.​1782 21.Wittwer, J.: A sharp estimate on the norm of the martingale transform. Math. Res. Lett. 7(1), 1–12 (2000)MathSciNetCrossRefMATH About this Article Title Sharp Estimates for Lipschitz Class Journal The Journal of Geometric Analysis Volume 26, Issue 2 , pp 1346-1369 Cover Date2016-04 DOI 10.1007/s12220-015-9593-7 Print ISSN 1050-6926 Online ISSN 1559-002X Publisher Springer US Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds Keywords Sharp function Lipschitz Best constants 26A16 Authors Adam Osȩkowski (1) Author Affiliations 1. Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland Continue reading... To view the rest of this content please follow the download PDF link above.