Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

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• 作者：Paul Hagelstein ; Ioannis Parissis
• 关键词：Halo function ; Muckenhoupt weights ; Doubling measure ; Maximal function ; Tauberian conditions
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：26
• 期：2
• 页码：924-946
• 全文大小：555 KB
• 参考文献：1.Beznosova, O.V., Hagelstein, P.A.: Continuity of halo functions associated to homothecy invariant density bases. Colloquium Mathematicum 134(2), 235–243 (2014)MathSciNet CrossRef MATH
  2.Beznosova, O., Reznikov, A.: Sharp estimates involving \(A_\infty \) and \(L\log L\) constants, and their applications to PDE. Algebra i Anal. 26(1), 40–67 (2014). MR3234812MathSciNet MATH
  3.Cabrelli, C., Lacey, M.T., Molter, U., Pipher, J.C.: Variations on the theme of Journé’s lemma. Houston J. Math. 32(3), 833–861 (2006). MR2247912 (2007e:42011)MathSciNet MATH
  4.Córdoba, A., Fefferman, R.: A geometric proof of the strong maximal theorem. Ann. Math. 102(1), 95–100 (1975). MR0379785 (52 #690)MathSciNet CrossRef MATH
  5.Córdoba, A., Fefferman, R.: On the equivalence between the boundedness of maximal and multiplier operators in Fourier analysis. Proc. Natl. Acad. Sci. USA 74(2), 423–425 (1977). MR0433117MathSciNet CrossRef MATH
  6.Dindoš, M., Wall, T.: The sharp \(A_p\) constant for weights in a reverse-Hölder class. Rev. Mat. Iberoam. 25(2), 559–594 (2009). MR2569547 (2011b:42041)MathSciNet CrossRef MATH
  7.Duoandikoetxea, J., Martín-Reyes, F.J., Ombrosi, S.: Calderón weights as Muckenhoupt weights. Indiana Univ. Math. J. 62(3), 891–910 (2013). MR3164849MathSciNet CrossRef MATH
  8.Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Japon. 22(5), 529–534 (1977/78). MR0481968 (58 #2058)
  9.Füredi, Z., Loeb, P.A.: On the best constant for the Besicovitch covering theorem. Proc. Am. Math. Soc. 121(4), 1063–1073 (1994). MR1249875 (95b:28003)MathSciNet CrossRef MATH
  10.García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studiesm, vol. 116. North-Holland Publishing Co., Amsterdam (1985). Notas de Matemática [Mathematical Notes], vol. 104 (1985) MR807149 (87d:42023)
  11.Garnett, J.B.: Bounded analytic functions. Graduate Texts in Mathematics, 1st edn. Springer, New York (2007). MR2261424 (2007e:30049)
  12.de Guzmán, M.: Differentiation of integrals in \({\bf R}^{n}\) . Measure Theory (Proc. Conf., Oberwolfach, 1975), Lecture Notes in Math., vol. 541, pp. 181–185. Springer, Berlin (1976). MR0476978 (57 #16523)
  13.Hagelstein, P. A., Luque, T., Parissis, I.: Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Trans. Am. Math. Soc. (To appear). arXiv:​1304.​1015
  14.Hagelstein, P.A., Parissis, I.: Solyanik estimates in harmonic analysis. In: Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol. 108, pp. 87–103. Springer, Heidelberg (2014)
  15.Hagelstein, P.A., Stokolos, A.: Tauberian conditions for geometric maximal operators. Trans. Am. Math. Soc. 361(6), 3031–3040 (2009). MR2485416 (2010b:42023)MathSciNet CrossRef MATH
  16.Hruščev, S.V.: A description of weights satisfying the \(A_{\infty }\) condition of Muckenhoupt. Proc. Am. Math. Soc. 90(2), 253–257 (1984). MR727244 (85k:42049)
  17.Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \) . Anal. PDE 6(4), 777–818 (2013). MR3092729MathSciNet CrossRef MATH
  18.Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous type. J. Funct. Anal. 263(12), 3883–3899 (2012). MR2990061MathSciNet CrossRef MATH
  19.Korey, M.B.: Ideal weights: asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation. J. Fourier Anal. Appl. 4(4–5), 491–519 (1998). MR1658636 (99m:42032)MathSciNet CrossRef MATH
  20.Lerner, A.K., Moen, K.: Mixed \(A_p\) -\(A_\infty \) estimates with one supremum. Studia Math. 219(3), 247–267 (2013). MR3145553MathSciNet CrossRef MATH
  21.Mitsis, T.: Embedding \(B_\infty \) into Muckenhoupt classes. Proc. Am. Math. Soc. 133(4), 1057–1061 (2005). (electronic) MR2117206 (2005i:42031)MathSciNet CrossRef MATH
  22.Politis, A.: Sharp results on the relation between weight spaces and BMO. ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.). The University of Chicago 1995. MR2716561
  23.Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973). MR0365062MATH
  24.Solyanik, A.A.: On halo functions for differentiation bases. Mat. Zametki, 54(6), 82–89, 160 (1993). (Russian, with Russian summary): English transl., Math. Notes, 54 (1993) (5–6), 1241–1245 (1994) MR1268374 (95g:42033)
  25.Wik, I.: On Muckenhoupt’s classes of weight functions. Studia Math. 94(3), 245–255 (1989). MR1019792 (90j:42029)MathSciNet MATH
  26.Wilson, J.: Michael weighted inequalities for the dyadic square function without dyadic \(A_\infty \) . Duke Math. J. 55(1), 19–50 (1987). MR883661 (88d:42034)MathSciNet CrossRef MATH
  27.Wilson, M.: Weighted Littlewood–Paley Theory and Exponential-Square Integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008). MR2359017 (2008m:42034)MATH
  
• 作者单位：Paul Hagelstein (1)
  Ioannis Parissis (2)
  
  1. Department of Mathematics, Baylor University, Waco, TX, 76798, USA
  2. Department of Mathematics, Aalto University, P. O. Box 11100, 00076, Espoo, Finland
  
• 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 denote a weight in which belongs to the Muckenhoupt class and let denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure . The sharp Tauberian constant of with respect to , denoted by , is defined by In this paper, we show that the Solyanik estimate $$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator and a weight : We show that we have if and only if . As a corollary of our methods we obtain a quantitative embedding of into . Keywords Halo function Muckenhoupt weights Doubling measure Maximal function Tauberian conditions Mathematics Subject Classification Primary 42B25 Secondary 42B35 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 (27) References1.Beznosova, O.V., Hagelstein, P.A.: Continuity of halo functions associated to homothecy invariant density bases. Colloquium Mathematicum 134(2), 235–243 (2014)MathSciNetCrossRefMATH2.Beznosova, O., Reznikov, A.: Sharp estimates involving \(A_\infty \) and \(L\log L\) constants, and their applications to PDE. Algebra i Anal. 26(1), 40–67 (2014). MR3234812MathSciNetMATH3.Cabrelli, C., Lacey, M.T., Molter, U., Pipher, J.C.: Variations on the theme of Journé’s lemma. Houston J. Math. 32(3), 833–861 (2006). MR2247912 (2007e:42011)MathSciNetMATH4.Córdoba, A., Fefferman, R.: A geometric proof of the strong maximal theorem. Ann. Math. 102(1), 95–100 (1975). MR0379785 (52 #690)MathSciNetCrossRefMATH5.Córdoba, A., Fefferman, R.: On the equivalence between the boundedness of maximal and multiplier operators in Fourier analysis. Proc. Natl. Acad. Sci. USA 74(2), 423–425 (1977). MR0433117MathSciNetCrossRefMATH6.Dindoš, M., Wall, T.: The sharp \(A_p\) constant for weights in a reverse-Hölder class. Rev. Mat. Iberoam. 25(2), 559–594 (2009). MR2569547 (2011b:42041)MathSciNetCrossRefMATH7.Duoandikoetxea, J., Martín-Reyes, F.J., Ombrosi, S.: Calderón weights as Muckenhoupt weights. Indiana Univ. Math. J. 62(3), 891–910 (2013). MR3164849MathSciNetCrossRefMATH8.Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Japon. 22(5), 529–534 (1977/78). MR0481968 (58 #2058)9.Füredi, Z., Loeb, P.A.: On the best constant for the Besicovitch covering theorem. Proc. Am. Math. Soc. 121(4), 1063–1073 (1994). MR1249875 (95b:28003)MathSciNetCrossRefMATH10.García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studiesm, vol. 116. North-Holland Publishing Co., Amsterdam (1985). Notas de Matemática [Mathematical Notes], vol. 104 (1985) MR807149 (87d:42023)11.Garnett, J.B.: Bounded analytic functions. Graduate Texts in Mathematics, 1st edn. Springer, New York (2007). MR2261424 (2007e:30049)12.de Guzmán, M.: Differentiation of integrals in \({\bf R}^{n}\). Measure Theory (Proc. Conf., Oberwolfach, 1975), Lecture Notes in Math., vol. 541, pp. 181–185. Springer, Berlin (1976). MR0476978 (57 #16523)13.Hagelstein, P. A., Luque, T., Parissis, I.: Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Trans. Am. Math. Soc. (To appear). arXiv:​1304.​1015 14.Hagelstein, P.A., Parissis, I.: Solyanik estimates in harmonic analysis. In: Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol. 108, pp. 87–103. Springer, Heidelberg (2014)15.Hagelstein, P.A., Stokolos, A.: Tauberian conditions for geometric maximal operators. Trans. Am. Math. Soc. 361(6), 3031–3040 (2009). MR2485416 (2010b:42023)MathSciNetCrossRefMATH16.Hruščev, S.V.: A description of weights satisfying the \(A_{\infty }\) condition of Muckenhoupt. Proc. Am. Math. Soc. 90(2), 253–257 (1984). MR727244 (85k:42049)17.Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013). MR3092729MathSciNetCrossRefMATH18.Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous type. J. Funct. Anal. 263(12), 3883–3899 (2012). MR2990061MathSciNetCrossRefMATH19.Korey, M.B.: Ideal weights: asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation. J. Fourier Anal. Appl. 4(4–5), 491–519 (1998). MR1658636 (99m:42032)MathSciNetCrossRefMATH20.Lerner, A.K., Moen, K.: Mixed \(A_p\)-\(A_\infty \) estimates with one supremum. Studia Math. 219(3), 247–267 (2013). MR3145553MathSciNetCrossRefMATH21.Mitsis, T.: Embedding \(B_\infty \) into Muckenhoupt classes. Proc. Am. Math. Soc. 133(4), 1057–1061 (2005). (electronic) MR2117206 (2005i:42031)MathSciNetCrossRefMATH22.Politis, A.: Sharp results on the relation between weight spaces and BMO. ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.). The University of Chicago 1995. MR271656123.Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973). MR0365062MATH24.Solyanik, A.A.: On halo functions for differentiation bases. Mat. Zametki, 54(6), 82–89, 160 (1993). (Russian, with Russian summary): English transl., Math. Notes, 54 (1993) (5–6), 1241–1245 (1994) MR1268374 (95g:42033)25.Wik, I.: On Muckenhoupt’s classes of weight functions. Studia Math. 94(3), 245–255 (1989). MR1019792 (90j:42029)MathSciNetMATH26.Wilson, J.: Michael weighted inequalities for the dyadic square function without dyadic \(A_\infty \). Duke Math. J. 55(1), 19–50 (1987). MR883661 (88d:42034)MathSciNetCrossRefMATH27.Wilson, M.: Weighted Littlewood–Paley Theory and Exponential-Square Integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008). MR2359017 (2008m:42034)MATH About this Article Title Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into Journal The Journal of Geometric Analysis Volume 26, Issue 2 , pp 924-946 Cover Date2016-04 DOI 10.1007/s12220-015-9578-6 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 Halo function Muckenhoupt weights Doubling measure Maximal function Tauberian conditions Primary 42B25 Secondary 42B35 Authors Paul Hagelstein (1) Ioannis Parissis (2) Author Affiliations 1. Department of Mathematics, Baylor University, Waco, TX, 76798, USA 2. Department of Mathematics, Aalto University, P. O. Box 11100, 00076, Espoo, Finland Continue reading... 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