Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 13 (2018), 41 -- 94

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

THE T(1) THEOREM REVISITED

Josefina Álvarez and Martha Guzmán-Partida

Abstract. The main purpose of this article is to present a proof of the T(1) Theorem that uses a continuous version of the Cotlar-Knapp-Stein lemma, due to A. P. Calderón and R. Vaillancourt.

2010 Mathematics Subject Classification: 42B20; 30H35; 42B30.
Keywords: Singular integral operators; weak boundedness property; BMO functions; Carleson measures.

Full text

References

  1. J. Álvarez, An algebra of Lp-bounded pseudo-differential operators, J. Math. Anal. and Appl. Vol. 94 No. 1 (1983), 268-282. MR0701463. Zbl 0519.35084.

  2. J. Álvarez and J. Hounie, Estimates for the kernel of pseudo-differential operators and applications, Ark. för Mat. 28 (1990), 1-22. MR1049640. Zbl 0713.35106.

  3. J. Álvarez, J. Hounie and C. Pérez, A pointwise estimate for the kernel of a pseudo-differential operator, with applications, Rev. Un. Mat. Argentina 37 no. 3-4 (1992), 184-199. MR1266682. Zbl 0798.35171.

  4. J. Álvarez and M. Milman, Hp continuity properties of Calderón-Zygmund type operators, J. Math. Anal. and Appl. Vol. 118 No. 1 (1986), 63-79. MR0849442. Zbl 0596.42006.

  5. ---------------------------------, Vector valued inequalities for strongly singular Calderón-Zygmund operators, Rev. Mat. Iberoamericana Vol. 2 No. 4 (1986), 405-426. MR0913696. Zbl 634.42016.

  6. J. Barros-Neto, An Introduction to the Theory of Distributions, Reprint Edition, Krieger Publishing Co. 1981. MR0461128. Zbl 051246040.

  7. R. Beals, A general calculus of pseudo-differential operators, Duke Math. J. 42 (1975), 1-42. MR0367730. Zbl 0343.35078.

  8. J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux derivées partielles nonlinéaires, Ann. Sc. École Norm. Sup. 4, 14 (1981), 209-246. MR0631715. Zbl 0495.35024.

  9. G. Bourdaud, Une algèbre maximale d'opératéurs pseudo-différentiels, Comm. Partial Differential Equations, 13 (1988), 1059-1083. MR0946282. Zbl 0659.35115.

  10. A. P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. MR0104925. Zbl 0080.30302.

  11. --------------------, Integrales singulares y sus aplicaciones a ecuaciones diferenciales hiperbólicas, Cursos y Seminarios de Matemática 3, 1960. Universidad de Buenos Aires. MR0123834. Zbl 0096.0672.

  12. --------------------, Existence and uniqueness theorems for systems of partial differential equations, Symposium on Fluid Dynamics, Institute for Fluid Dynamics, University of Maryland, College Park. 1961. MR0156078. Zbl 0147.08202.

  13. --------------------, Algebras of singular integral operators, Proc. Symp. Pure Math. 10, 1965. Amer. Math. Soc. MR0394309. Zbl 0182.47402.

  14. --------------------, Commutators of singular integrals operators, Proc. Nat. Acad. Sci. USA 53 (1965), 1092-1099. MR0177312. Zbl 0151.16901.

  15. ------------------, Singular integrals, Bulletin Amer. Math. Soc. 72 (1966), 427-465. MR0209918. Zbl 0151.16902.

  16. --------------------, A priori estimates for singular integral operator, Pseudo-Differential Operators, CIME, Stresa, Edizioni Cremonese, Rome. (1969), 85-141. MR0256215.

  17. ------------------, Lectures notes on pseudo-differential operators and elliptic boundary value problems, I, Cursos de Matemática I, 1976 Instituto Argentino de Matemática. MR0460947. Zbl 0335.35082.

  18. ------------------, Cauchy integrals on Lipschitz curves and related topics, Proc. Nat. Acad. Sci. USA 74 (1977), 1324-1327. MR0466568. Zbl 0373.44003.

  19. ------------------, Commutators, singular integrals on Lipschitz curves and applications, Proc. of the ICM, Helsinski,1978. MR0562599. Zbl 0429.35077.

  20. A. P. Calderón and R. Vaillancourt, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci. USA 69 (1972), 1185-1187. MR0298480. Zbl 0244.35074.

  21. A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. MR0052553. Zbl 0047.10201.

  22. ------------------------------------------, On singular integrals, Amer. J. Math. 78 (1956), 310-320. MR0084633. Zbl 0072.11501.

  23. ------------------------------------------, Algebras of certain singular operators, Amer. J. Math. 78 (1956), 310-320. MR0087810. Zbl 0072.11601.

  24. C.-H. Ching, Pseudo-differential operators with non-regular symbols, J. Diff. Eq. Vol. 11 (1972), 436-447. MR0296764. Zbl 0248.35106.

  25. M. Christ, Lectures on Singular Integral Operators, CBMS Regional Conference Series in Mathematics 77, 1990 Amer. Math. Soc. MR1104656. Zbl 0745.42008.

  26. M. Christ and J.-L. Journé, Polynomial growth estimates for multilinear singular integrals, Acta Math. 159 (1987), 51-80. MR0906525. Zbl 0645.42017.

  27. R. R. Coifman, G. David and Y. Meyer, La solution des conjectures de Calderón, Adv. in Math. 48 (1983), 144-148. MR0700980. Zbl 0518.42024.

  28. R. R. Coifman, A. McIntosh and Y. Meyer, L'integrale de Cauchy définit un opératéur borné sur L2 pour les courbes lipschitziennes, Ann. of Math. 116 (1982), 361-388. MR0672839. Zbl 0497.42012.

  29. R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc.  212 (1975), 425-428. MR0380244. Zbl 0324.44005.

  30. --------------------------------------, Au Delà des Opérateurs Pseudo-Différentiels, Astérisque 57, 1978. Soc. Math. France. MR0518170. Zbl 0483.35082.

  31. --------------------------------------, Le theorème de Calderón par les méthodes de variable reélle, C. R. Acad.Sc. Paris 289 (1979), 425-428. MR0554634. Zbl 0427.42007.

  32. --------------------------------------, Fourier analysis of multilinear convolutions, Calderón's theorem and analysis on Lipschitz curves, Lecture Notes in Math. , Springer-Verlag  779 (1979), 109-122. MR0576041. Zbl 0427.42006.

  33. --------------------------------------, A simple proof of a theorem by G. David and J.-L. Journé on singular integral operators, Probability Theory and Harmonic Analysis, (1986), 61-65. Marcel Dekker. MR083023. Zbl 0603.42018.

  34. --------------------------------------, Ondelettes et Opératéurs, Vol. III, Hermann 1991. MR1160989. Zbl 0745.42012.

  35. M. Cotlar, A combinatorial inequality and its application to L2 spaces, Rev. Mat. Cuyana, Vol. 1 (1955), 41-55. MR0080263. Zbl 0071.33301.

  36. G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sc. École Norm. Sup. 17 (1984), 157-189. MR0744071. Zbl 0537.42016.

  37. ------------, Wavelets and Singular Integrals on Curves, Lectures Notes in Math. 1465, 1991. Springer-Verlag. MR1123480. Zbl 0764.42019.

  38. G. David and J.-L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. 120 No. 2 (1984), 371-397. MR0763911. Zbl 0567.47025.

  39. G. David, J.-L. Journé and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), 1-56. MR0850408. Zbl 0604.42014.

  40. J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, Vol. 29, 2001. Amer. Math. Soc. MR1800316. Zbl 0969.42001.

  41. E. Fabes, I. Mitrea and M. Mitrea, On the boundedness of singular integrals, Pacific J. Math. 189 No. 1 (1999), 21-29. MR1687806. Zbl 1054.42503.

  42. C. Fefferman, Lp bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413-417. MR0336453. Zbl 0259.47045.

  43. L. Grafakos, Classical Fourier Analysis, Third Edition, Springer 2014. MR3243734. Zbl 0259.47045.

  44. ---------------, Modern Fourier Analysis, Third Edition, Springer 2014. MR3243741. Zbl 1304.42002.

  45. L. Grafakos and R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. in Math. .165 no. 1, 2002. MR1880324. Zbl 1032.42020.

  46. J. Hart, A new proof of the bilinear T(1) Theorem, Proc. Amer. Math. Soc. 142 no. 9 (2014), 3169-3181. MR3223373. Zbl 06345067.

  47. J. Heinonen, Lectures on Lipschitz Analysis, 14th Jyväskylä Summer School, August 2004. MR2177410. Zbl 1086.30003. www.math.jyu.fi/research/repors/rep100.pdf.

  48. L. Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501-517. MR0180740. Zbl 0125.33401.

  49. ------------------, Pseudo-differential operators and non elliptic boundary problems, Ann. of Math. 83 (1966), 129-209. MR0233064. Zbl 0132.07402.

  50. ------------------, Pseudo-differential operators and hypoelliptic equations, Proc. Symp. Pure Math. 10 (1966), 138-183. MR0383152. Zbl 0167.09603.

  51. ------------------, On the L2 continuity of pseudo-differential operators, Comm. Pure Appl. Math. 24 (1971), 529-535. MR0281060. Zbl 0206.39303.

  52. ------------------, The Analysis of Linear Partial Differential Operators, I-IV, Springer 1984. MR0717035. Zbl 1028.35001.

  53. ------------------, Pseudo-differential operators of type (1,1), Comm. Partial Differential Equations 13(1988), 1085-1111. MR0946283. Zbl 0667.35078.

  54. ------------------, Continuity of pseudo-differential operators of type (1,1), Comm. Partial Differential Equations 14 (1989), 231-243. MR0976972. Zbl 0688.35107.

  55. J. Hounie, On the L2 continuity of pseudo-differential operators, Comm. Partial Differential Equations 11 (1986), 765-778. MR0837930. Zbl 0597.35121.

  56. -----------, Local solvability of first order linear operators with Lipschitz coefficients, Duke Math. J. 62 no.2 (1991), 467-477. MR1104533. Zbl 0731.35025.

  57. J.-L. Journé, Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math.994, 1983. Springer-Verlag. MR0706075. Zbl 0508.42021.

  58. A. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. 93 (1971), 489-578. MR0460543. Zbl 0257.22015.

  59. J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269-305. MR0176362. Zbl 0171.35101.

  60. H. Kumano-Go, Algebras of pseudo-differential operators, J. Fac. Sci. Univ. Tokyo, 17 (1970), 31-50. MR0291896. Zbl 0206.10501.

  61. Y. Meyer, The paradifferential operators, preprint, 1979.

  62. ------------, Les Nouveaux Opérateurs de Calderón-Zygmund, Astérisque 131 (1985), 237-254. Soc. Math. France. MR0816750. Zbl 0573.42010.

  63. ------------, Real analysis and operator theory, Proc. Symp. Pure Math. 43 (1985), 219-235. MR0812293. Zbl 0591.47041.

  64. Y. Meyer and Q. Yang, Continuity of Calderón-Zygmund operators, http://www.cmap.polytechnique.fr/edp-cs/pdf/article_Y_Meyer.pdf.

  65. ------------------------------, Continuity of Calderón-Zygmund operators on Besov or Triebel-Lizorkin spaces, Anal. Appl. (Singap.) 6 No. 1 (2008), 51-81. MR2380886. Zbl 1268.42026.

  66. N. Miller, Weighted Sobolev spaces and pseudo-differential operators with smooth symbols, Trans. Amer. Math. Soc. 969 (1982), 91-109. MR0637030. Zbl 0482.35082.

  67. T. Murai, Boundedness of singular integral operators of Calderón type, V, Adv. in Math. 59 (1986), 71-81. MR0825088. Zbl 0608.42010.

  68. T. Murai, A Real-Variable Method for the Cauchy Transform and Analytic Capacity, Lecture Notes in Math. 1307, 1988. Springer-Verlag. MR0944308. Zbl 0645.30016.

  69. N. L. Muskhelishvili, Singular Integral Equations, English translation, J. R. M. Radok (Ed.), Wolters-Noordhoff 1972. MR0355494. Zbl 0488.45002.

  70. M. Nagase, The Lp boundedness of pseudo-differential operators with non regular symbols, Comm. Partial Differential Equations, 2 (1977), 1045-1061. MR0470758. Zbl 0397.35071.

  71. A. Nagel and E. M. Stein, Lectures on Pseudo-Differential Operators: Regularity Theorems and Applications to Non Elliptic Problems, Math. Notes 24, 1979. Princeton University Press. MR0549321. Zbl 0415.47025.

  72. J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 67 (1986), 7-48. MR0859252. Zbl 0627.42008.

  73. L. Schwartz, Théorie des noyaux, Proc. of the ICM (Cambridge, Mass. 1950) Vol. 1, Providence, R. I. Amer. Math. Soc. (1952), 220-230. MR0045307. Zbl 0048.35102.

  74. R. Seeley, Refinement of the functional calculus of Calderón and Zygmund, Proc. Dutch Acad. Ser. A 68 (1965), 521-531. MR0226450. Zbl 0141.13302.

  75. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. MR0290095. Zbl 0207.13501.

  76. ---------------, Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, 307-355, 1986. Annals of Mathematics Studies, Princeton University Press. MR0864375. Zbl 0618.42006.

  77. ---------------, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press 1993. MR12322192. Zbl 0821.42001.

  78. E. M. Stein and G. Weiss, Introduction to Fourier Analysis in Euclidean Spaces. Princeton University Press 1971. MR0304972. Zbl 0232.42007.

  79. R. S. Strichartz, Paradifferential operators: Another step forward for the method of Fourier, Notices AMS 29 (1982), 401-406.

  80. T. Tao, The T(b) Theorem and its variants, Proc. Centre Mathematics and Its Applications 41, 2003. 143-160, Mathematical Sciences Institute, The Australian National University, Camberra. MR1994522.

  81. ---------, Distinguished Lecture Series II: Elias Stein, \textquotedblleft The Cauchy Integral in ℂn\textquotedblright , https://terrytao.wordpress.com/2008/10/23/distinguished-lecture-series-ii-elias-stein-the-cauchy-integral.

  82. ---------, Lecture Notes 6 for 247B (Paraproducts) https://www.math.ucla.edu/~tao/247b.1.07w/notes6.pdf.

  83. ---------, Lecture Notes 7 for 247B (The T(1) Thm.) https://www.math.ucla.edu/~tao/247b.1.07w/notes7.pdf.

  84. M. Taylor, Pseudo-Differential Operators, Princeton University Press 1981. MR0618463. Zbl 0289.35001.

  85. X. Tolsa, Littlewood-Paley theory and the T(1) Theorem with non-doubling measures, Adv. in Math. 164 Issue 1 (2001) 57-116. MR1870513. Zbl 1015.42010.

  86. R. H. Torres, Boundedness Results for Operators with Singular Kernels on Distribution Spaces, Memoirs of the Amer. Math. Soc. no. 442, 1992. MR1048075.

  87. A. Unterberger and J. Bokobza, Les opératéurs de Calderón-Zygmund precisés, C. R. Acad. Sc. Paris 259 (1964), 1612-1614. MR0176360. Zbl 0143.36904.

  88. ------------------------------------------, Sur une généralisation des opérateurs de Calderón-Zygmund et des spaces Hs, C. R. Acad. Sc. Paris 260 (1965), 3265-3267. MR0188832. Zbl 0143.37002.

  89. P. S. Viola, Vector valued T(1) Theorem and Littlewood-Paley theory on spaces of homogeneous type, Rev. Un. Mat. Argentina 55 No. 1 (2014), 25-53. MR3264683. Zbl 1334.42035.

  90. K. Wang, The generalization of paraproducts and the full T1 Theorem for Sobolev and Triebel-Lizorkin spaces, J. Math. Anal. and Appl. 209 Issue 2 (1997), 317-340. MR1474612. Zbl 0886.46040.

  91. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press 1945. MR0178117. Zbl 0951.30002.







Josefina Alvarez
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003, USA.
jalvarez@nmsu.edu


Martha Guzmán-Partida
Departamento de Matemáticas
Universidad de Sonora
Hermosillo, Sonora, 83000, México.
martha@mat.uson.mx

http://www.utgjiu.ro/math/sma