$L^p$ theory for outer measures and two themes of Lennart Carleson united
HTML articles powered by AMS MathViewer
- by Yen Do and Christoph Thiele PDF
- Bull. Amer. Math. Soc. 52 (2015), 249-296 Request permission
Abstract:
We develop a theory of $L^p$ spaces based on outer measures generated through coverings by distinguished sets. The theory includes as a special case the classical $L^p$ theory on Euclidean spaces as well as some previously considered generalizations. The theory is a framework to describe aspects of singular integral theory, such as Carleson embedding theorems, paraproduct estimates, and $T(1)$ theorems. It is particularly useful for generalizations of singular integral theory in time-frequency analysis, the latter originating in Carleson’s investigation of convergence of Fourier series. We formulate and prove a generalized Carleson embedding theorem and give a relatively short reduction of the most basic $L^p$ estimates for the bilinear Hilbert transform to this new Carleson embedding theorem.References
- David R. Adams, Choquet integrals in potential theory, Publ. Mat. 42 (1998), no. 1, 3–66. MR 1628134, DOI 10.5565/PUBLMAT_{4}2198_{0}1
- Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
- Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. MR 199631, DOI 10.1007/BF02392815
- R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
- Guy David and Jean-Lin Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 763911, DOI 10.2307/2006946
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- Ciprian Demeter and Christoph Thiele, On the two-dimensional bilinear Hilbert transform, Amer. J. Math. 132 (2010), no. 1, 201–256. MR 2597511, DOI 10.1353/ajm.0.0101
- Yen Do, Camil Muscalu, and Christoph Thiele, Variational estimates for paraproducts, Rev. Mat. Iberoam. 28 (2012), no. 3, 857–878. MR 2949622, DOI 10.4171/RMI/694
- Loukas Grafakos and Xiaochun Li, Uniform bounds for the bilinear Hilbert transforms. I, Ann. of Math. (2) 159 (2004), no. 3, 889–933. MR 2113017, DOI 10.4007/annals.2004.159.889
- Michael Lacey and Christoph Thiele, $L^p$ estimates for the bilinear Hilbert transform, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 1, 33–35. MR 1425870, DOI 10.1073/pnas.94.1.33
- Michael T. Lacey and Christoph M. Thiele, On Calderón’s conjecture for the bilinear Hilbert transform, Proc. Natl. Acad. Sci. USA 95 (1998), no. 9, 4828–4830. MR 1619285, DOI 10.1073/pnas.95.9.4828
- Michael Lacey and Christoph Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7 (2000), no. 4, 361–370. MR 1783613, DOI 10.4310/MRL.2000.v7.n4.a1
- Andrei K. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141–161. MR 3127380, DOI 10.1007/s11854-013-0030-1
- Camil Muscalu, Terence Tao, and Christoph Thiele, $L^p$ estimates for the biest. II. The Fourier case, Math. Ann. 329 (2004), no. 3, 427–461. MR 2127985, DOI 10.1007/s00208-003-0508-8
- Camil Muscalu, Terence Tao, and Christoph Thiele, Uniform estimates on multi-linear operators with modulation symmetry, J. Anal. Math. 88 (2002), 255–309. Dedicated to the memory of Tom Wolff. MR 1979774, DOI 10.1007/BF02786579
- Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, and James Wright, A variation norm Carleson theorem, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 421–464. MR 2881301, DOI 10.4171/JEMS/307
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Christoph Thiele, The quartile operator and pointwise convergence of Walsh series, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5745–5766. MR 1695038, DOI 10.1090/S0002-9947-00-02577-0
- Christoph Thiele, A uniform estimate, Ann. of Math. (2) 156 (2002), no. 2, 519–563. MR 1933076, DOI 10.2307/3597197
- Richard L. Wheeden and Antoni Zygmund, Measure and integral, Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977. An introduction to real analysis. MR 0492146, DOI 10.1201/b15702
Additional Information
- Yen Do
- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
- MR Author ID: 940906
- Email: yen.do@yale.edu
- Christoph Thiele
- Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Alle 60, D-53115 Bonn, and Department of Mathematics, UCLA, Los Angeles, California 90095
- Email: thiele@math.uni-bonn.de
- Received by editor(s): September 4, 2013
- Published electronically: December 29, 2014
- Additional Notes: The first author was partially supported by NSF grant DMS 1201456
The second author was partially supported by NSF grant DMS 1001535. - © Copyright 2014 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 52 (2015), 249-296
- MSC (2010): Primary 42B20
- DOI: https://doi.org/10.1090/S0273-0979-2014-01474-0
- MathSciNet review: 3312633
Dedicated: Dedicated to Lennart Carleson