Vanishing Carleson Measures Associated with Families of Multilinear Operators

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• 作者：Yong Ding ; Ting Mei
• 关键词：Vanishing Carleson measure ; Multilinear operator ; Paraproduct ; Compactness ; CMO(\(\mathbb {R}^{n}\) ; )
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：26
• 期：2
• 页码：1539-1559
• 全文大小：538 KB
• 参考文献：1.Auscher, P., McIntosh, A., Hofmann, S., Lacey, M., Tchamitchian, P.: The solution of the Kato square root problem for second order elliptic operators on \(\mathbb{R}^n\) . Ann. Math. 156, 633–654 (2002)MathSciNet CrossRef MATH
  2.Carleson, L.: An interpolation problem for bounded analytic functions. Am. J. Math. 80, 921–930 (1958)MathSciNet CrossRef MATH
  3.Carleson, L.: Interpolation by bounded analytic functions and the corona problem. Ann. Math. 76, 547–559 (1962)MathSciNet CrossRef MATH
  4.Christ, M.: Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77. American Mathematical Society, Providence (1990)
  5.Coifman, R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur \(L^2\) pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)MathSciNet CrossRef MATH
  6.Coifman, R., Meyer, Y.: Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves, Euclidean harmonic analysis (Proceedings in SEM University of Maryland, College Park, MD., 1979). Lecture Notes in Mathematics, vol. 779, pp. 104–122. Springer, Berlin (1980)
  7.Coifman, R., Meyer, Y.: In: Chao, J., Woyczyński, W. (eds.) Probability Theory and Harmonic Analysis. A simple proof of a theorem by G. David and J.-L. Journé on singular integral operators, pp. 61–65. Marcel Dekker, New York (1986)
  8.Colzani, L.: Hardy Spaces on Sphere. Ph.D. Thesis, Washington University, St. Louis (1982)
  9.de Guzman, M.: Real-Variable Methods in Fourier Analysis. North-Holland mathematics studies. North-Holland, Amsterdam (1981)MATH
  10.Ding, Y., Fan, D., Pan, Y.: \(L^p\) -boundedness of Marcinkiewicz integrals with Hardy space function kernels. Acta Math. Sinica 16, 593–600 (2000)MathSciNet CrossRef MATH
  11.Ding, Y., Mei, T.: Some characterizations of \(VMO(\mathbb{R}^n)\) . Anal. Theory Appl. 30, 387–398 (2014)MathSciNet MATH
  12.Ding, Y., Xiao, Y.: Carleson measure with rough kernels. Chin. Math. Ann. 29(A(6)), 801–808 (2008). (in Chinese)MathSciNet MATH
  13.Duoandikoetxea, J.: Fourier Analysis. Graduate sudies in mathematics, 29th edn. American Mathematical Society, Providence (2001)MATH
  14.Fan, D., Pan, Y.: Singular integral operators with rough kernels supported by subvarieties. Am. J. Math. 119, 799–839 (1997)MathSciNet CrossRef MATH
  15.Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, 3rd edn. Springer, New York (2014)
  16.Grafakos, L., Oliveira, L.: Carleson measures associated with families of multilinear operators. Studia Math. 211, 71–94 (2012)MathSciNet CrossRef MATH
  17.Grafakos, L., Oliveira, L.: Corrigendum to “Carleson measures associated with families of multilinear operators”. Studia Math. 221, 193–196 (2014)MathSciNet CrossRef MATH
  18.Neri, U.: Fractional integration on the space \(H^1\) and its dual. Studia Math. 53, 175–189 (1975)MathSciNet MATH
  19.Rudin, W.: Functional Analysis. International series in pure and applied mathematics, 2nd edn. McGraw-Hill Inc., New York (1991)MATH
  20.Stein, E.M.: On the function of Littlewood-Paley, Lusin and Marcinkiewicz. Trans. Am. Math. Soc. 88, 430–466 (1958)CrossRef MATH
  21.Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillation Integrals. Princeton University Press, Princeton (1993)MATH
  22.Uchiyama, A.: On the compactness of operators of Hankel type. Tohoku Math. J. 30(2), 163–171 (1978)MathSciNet CrossRef MATH
  23.Walsh, T.: On the function of Marcinkiewicz. Studia Math. 44, 203–217 (1972)MathSciNet MATH
  
• 作者单位：Yong Ding (1)
  Ting Mei (1)
  
  1. Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China
  
• 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

In this paper we define a kind of vanishing Carleson measure on \(\mathbb {R}^{n+1}_+\) and give its characterization by the compact property of some convolution operator. We also investigate the construction of vanishing Carleson measures generated by a family of the multilinear operators \(\{\Theta _t\}_{t>0}\) and \(CMO\) functions. As some applications of our results, we also give the boundedness and compactness for the paraproduct \(\pi _{\vec {b}}\) associated with the family \(\{\Theta _t\}_{t>0}\) on \(L^2(\mathbb {R}^n)\), which is defined by $$\begin{aligned} \pi _{\vec {b}}(f)(x)= \int _0^\infty \eta _t*\big ((\varphi _t*f)\Theta _t(b_1,\ldots ,b_m)\big )(x)\; \frac{dt}{t}. \end{aligned}$$Further, for the linear case (i.e., \(m=1\)), we show that the paraproduct $$\begin{aligned} B_b(f)(x)=\int _0^\infty (f*\varphi _t)(x)(b*\psi _t)(x)\frac{\alpha (t)}{t}dt, \end{aligned}$$which was introduced by Coifman and Meyer, is also a compact operator on \(L^2(\mathbb {R}^n)\) if \(b\in CMO(\mathbb {R}^n)\) and \(\alpha \in L^\infty (\mathbb {R}^n)\). Keywords Vanishing Carleson measure Multilinear operator Paraproduct Compactness CMO(\(\mathbb {R}^{n}\)) Mathematics Subject Classification Primary 42B20 Secondary 42B25 47G10 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 (23) References1.Auscher, P., McIntosh, A., Hofmann, S., Lacey, M., Tchamitchian, P.: The solution of the Kato square root problem for second order elliptic operators on \(\mathbb{R}^n\). Ann. Math. 156, 633–654 (2002)MathSciNetCrossRefMATH2.Carleson, L.: An interpolation problem for bounded analytic functions. Am. J. Math. 80, 921–930 (1958)MathSciNetCrossRefMATH3.Carleson, L.: Interpolation by bounded analytic functions and the corona problem. Ann. Math. 76, 547–559 (1962)MathSciNetCrossRefMATH4.Christ, M.: Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77. American Mathematical Society, Providence (1990)5.Coifman, R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur \(L^2\) pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)MathSciNetCrossRefMATH6.Coifman, R., Meyer, Y.: Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves, Euclidean harmonic analysis (Proceedings in SEM University of Maryland, College Park, MD., 1979). Lecture Notes in Mathematics, vol. 779, pp. 104–122. Springer, Berlin (1980)7.Coifman, R., Meyer, Y.: In: Chao, J., Woyczyński, W. (eds.) Probability Theory and Harmonic Analysis. A simple proof of a theorem by G. David and J.-L. Journé on singular integral operators, pp. 61–65. Marcel Dekker, New York (1986)8.Colzani, L.: Hardy Spaces on Sphere. Ph.D. Thesis, Washington University, St. Louis (1982)9.de Guzman, M.: Real-Variable Methods in Fourier Analysis. North-Holland mathematics studies. North-Holland, Amsterdam (1981)MATH10.Ding, Y., Fan, D., Pan, Y.: \(L^p\)-boundedness of Marcinkiewicz integrals with Hardy space function kernels. Acta Math. Sinica 16, 593–600 (2000)MathSciNetCrossRefMATH11.Ding, Y., Mei, T.: Some characterizations of \(VMO(\mathbb{R}^n)\). Anal. Theory Appl. 30, 387–398 (2014)MathSciNetMATH12.Ding, Y., Xiao, Y.: Carleson measure with rough kernels. Chin. Math. Ann. 29(A(6)), 801–808 (2008). (in Chinese)MathSciNetMATH13.Duoandikoetxea, J.: Fourier Analysis. Graduate sudies in mathematics, 29th edn. American Mathematical Society, Providence (2001)MATH14.Fan, D., Pan, Y.: Singular integral operators with rough kernels supported by subvarieties. Am. J. Math. 119, 799–839 (1997)MathSciNetCrossRefMATH15.Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, 3rd edn. Springer, New York (2014)16.Grafakos, L., Oliveira, L.: Carleson measures associated with families of multilinear operators. Studia Math. 211, 71–94 (2012)MathSciNetCrossRefMATH17.Grafakos, L., Oliveira, L.: Corrigendum to “Carleson measures associated with families of multilinear operators”. Studia Math. 221, 193–196 (2014)MathSciNetCrossRefMATH18.Neri, U.: Fractional integration on the space \(H^1\) and its dual. Studia Math. 53, 175–189 (1975)MathSciNetMATH19.Rudin, W.: Functional Analysis. International series in pure and applied mathematics, 2nd edn. McGraw-Hill Inc., New York (1991)MATH20.Stein, E.M.: On the function of Littlewood-Paley, Lusin and Marcinkiewicz. Trans. Am. Math. Soc. 88, 430–466 (1958)CrossRefMATH21.Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillation Integrals. Princeton University Press, Princeton (1993)MATH22.Uchiyama, A.: On the compactness of operators of Hankel type. Tohoku Math. J. 30(2), 163–171 (1978)MathSciNetCrossRefMATH23.Walsh, T.: On the function of Marcinkiewicz. Studia Math. 44, 203–217 (1972)MathSciNetMATH About this Article Title Vanishing Carleson Measures Associated with Families of Multilinear Operators Journal The Journal of Geometric Analysis Volume 26, Issue 2 , pp 1539-1559 Cover Date2016-04 DOI 10.1007/s12220-015-9599-1 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 Vanishing Carleson measure Multilinear operator Paraproduct Compactness CMO( $$\mathbb {R}^{n}$$ R n ) Primary 42B20 Secondary 42B25 47G10 Authors Yong Ding (1) Ting Mei (1) Author Affiliations 1. Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China Continue reading... To view the rest of this content please follow the download PDF link above.