Abstract
We present a survey of certain aspects of the theory of singular integrals and square functions, with emphasis on L2 boundedness criteria and recent applications in partial differential equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Auscher P.: Regularity theorems and heat kernel for elliptic operators. J. Lond. Math. Soc. (2) 54(2), 284–296 (1996)
Auscher, P.: Lectures on the Kato square root problem. Surveys in analysis and operator theory (Canberra, 2001), pp. 1–18. Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40. Austral. Nat. Univ., Canberra (2002)
Auscher, P.: On necessary and sufficient conditions for Lp-estimates of Riesz transforms associated with elliptic operators on \({\mathbb {R}^n}\) and related estimates. Mem. Am. Math. Soc. 186, 871, xviii+75 (2007)
Alfonseca M., Auscher P., Axelsson A., Hofmann S., Kim S.: Analyticity of layer potentials and L2 solvability of boundary value problems for divergence form elliptic equations with complex L∞ coefficients. Adv. Math. 226(5), 4533–4606 (2011)
Auscher P., Axelsson A., Hofmann S.: Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems. J. Funct. Anal. 255, 374–448 (2008)
Auscher P., Axelsson A., McIntosh A.: Solvability of elliptic systems with square integrable boundary data. Ark. Mat. 48, 253–287 (2010)
Auscher P., Axelsson A., McIntosh A.: On a quadratic estimate related to the Kato conjecture and boundary value problems. Contemp. Math. AMS 505, 105–129 (2010)
Albrecht, D., Duong, X., McIntosh, A.: Operator theory and harmonic analysis. Workshop on Analysis and Geomerty 1995, Part III. Proc. of the CMA, ANU, Canberra, vol. 34, pp. 77–136 (1996)
Auscher P., Hofmann S., Lacey M., McIntosh A., Tchamitchian Ph.: The solution of the Kato square root problem for second order elliptic operators on \({\mathbb{R}^n}\) . Ann. Math. 156, 633–654 (2002)
Auscher P., Hofmann S., Lewis J.L., Tchamitchian Ph.: Extrapolation of Carleson measures and the analyticity of Kato’s square root operators. Acta Math. 187(2), 161–190 (2001)
Auscher, P., Hofmann, S., Martell, J.M.: Vertical versus conical square functions. Preprint
Auscher P., Hofmann S., McIntosh A., Tchamitchian Ph.: The Kato square root problem for higher order elliptic operators and systems on \({\mathbb{R}^n}\) . J. Evol. Equ. 1(4), 361–385 (2001)
Auscher P., Hofmann S., Muscalu C., Tao T., Thiele C.: Carleson measures, trees, extrapolation, and T (b) theorems. Publ. Mat. 46(2), 257–325 (2002)
Axelsson A., Keith S., McIntosh A.: Quadratic estimates and functional calculi of perturbed dirac operators. Invent. Math. 163, 455–497 (2006)
Axelsson A., Keith S., McIntosh A.: The Kato square root problem for mixed boundary value problems. J. Lond. Math. Soc. 74(2), 113–130 (2006)
Auscher P., McIntosh A., Tchamitchian Ph.: Heat kernels of second order complex elliptic operators and applications. J. Funct. Anal. 152, 22–73 (1998)
Auscher, P., Routin, E.: Local Tb theorems and Hardy inequalities. J. Geom. Anal., Online: doi:10.1007/s12220-011-9249-1
Auscher, P., Tchamitchian, Ph.: Square root problem for divergence operators and related topics. Astérisque 249. Société Mathématique de France (1998)
Auscher P., Yang Q.X.: On local T (b) theorems. Publ. Math. 53, 179–196 (2009)
Blunck S., Kunstmann P.: Calderón-Zygmund theory for non-integral operators and the H∞ functional calculus. Rev. Mat. Iberoamericana 19(3), 919–942 (2003)
Blunck S., Kunstmann P.: Weak type (p, p) estimates for Riesz transforms. Math. Z. 247, 137–148 (2004)
Bourgain J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21, 163–168 (1983)
Burkholder, D.L.: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. Conference on harmonic analysis in honor of Antoni Zygmund, vol. I, II (Chicago, Ill., 1981). Wadsworth Math. Ser., Wadsworth, Belmont, CA, pp. 270–286 (1983)
Burkholder, D.L.: Martingales and singular integrals in Banach spaces. Handbook of the Geometry of Banach Spaces, vol. I, pp. 233–269. North-Holland, Amsterdam (2001)
Calderón A.P.: Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 53, 1092–1099 (1965)
Calderón, A.P.: Algebras of singular integral operators. Proc. Sympos Pure Math., vol. 10, pp. 18–55. AMS, Providence (1967)
Calderón A.P.: Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA 74, 1324–1327 (1977)
Calderón, A.P.: Commutators, singular integrals on Lipschitz curves and applications. In: Proc. of the International Congress of Mathematicians (Helsinki, 1978), pp. 85–96. Acad. Sci. Fennica, Helsinki (1980)
Carleson L.: Interpolation of bounded analytic functions and the corona problem. Ann. Math. 76, 547–559 (1962)
Coifman R., Deng D., Meyer Y.: Domaine de la racine carrée de certains opérateurs différentiels accrétifs. Ann. Inst. Fourier (Grenoble) 33(2), 123–134 (1983)
Christ, M.: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloquium Mathematicum LX/LXI, pp. 601–628 (1990)
Christ, M.: Lectures on singular integral operators. CBMS Regional Conference Series in Mathematics, 77. Published for the Conference Board of the Mathematical Sciences, Washington, DC. Amer. Math. Soc., Providence (1990)
Christ M., Journé J.-L.: Polynomial growth estimates for multilinear singular integral operators. Acta Math. 159(1–2), 51–80 (1987)
Coifman R., Jones P., Semmes S.: Two elementary proofs of the L2 boundedness of Cauchy integrals on Lipschitz curves. J. Am. Math. Soc. 2(3), 553–564 (1989)
Coifman R., Meyer Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)
Coifman R., Meyer Y.: Commutateurs d’intégrales singuliéres et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28(3), 177–202 (1978)
Coifman, R., Meyer, Y.: Au delà des opérateurs pseudo differentiels, Astérisque. Soc. Math. 57 (1978)
Coifman, R., Meyer, Y.: Non-linear harmonic analysis and PDE. In: Stein, E.M. (ed.) Beijing Lectures in Harmonic Analysis. Annals of Math. Studies, vol. 112. Princeton University Press (1986)
Coifman R., McIntosh A., Meyer Y.: L’intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)
Calderón A.P., Zygmund A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)
David G.: Unrectifiable 1-sets have vanishing analytic capacity. Rev. Mat. Iberoamericana 14(2), 369–479 (1998)
David G.: Analytic capacity, Calderón-Zygmund operators, and rectifiability. Publ. Mat. 43(1), 3–25 (1999)
De Giorgi E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. (Italian). Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3), 25–43 (1957)
David G., Journé J.-L.: A boundedness criterion for generalized Calderón-Zygmund operators. Ann. Math. 120, 371–398 (1984)
David G., Journé J.-L., Semmes S.: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Mat. Iberoamericana 1, 1–56 (1985)
Duong X.T., McIntosh A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoamericana 15, 233–265 (1999)
Duong, X.T., McIntosh, A.: The Lp boundedness of Riesz transforms associated with divergence form operators. In: Workshop on Analysis and Applications, Brisbane, 1997. Proc. of the CMA, ANU, Canberra, vol. 37, pp. 15–25 (1999)
Figiel, T.: Singular integral operators: a martingale approach. Geometry of Banach spaces (Strobl, 1989). In: Müller, P.F.X., Schachermayer, W. (eds.) London Math. Soc. Lecture Note Ser., vol. 158, pp. 95–110. Cambridge University Press, Cambridge (1990)
Fabes E., Jerison D., Kenig C.: Multilinear Littlewood-Paley estimates with applications to partial differential equations . Proc. Natl. Acad. Sci. USA 79(18), 5746–5750 (1982)
Fabes E., Jodeit M., Rivière N.: Potential techniques for boundary value problems on C1-domains. Acta Math. 141, 165–186 (1978)
Fefferman C., Stein E.M.: Hp spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
Grau de la Herran, A., Mourgoglou, M.: Local Tb theorems for square functions on Ahlfors-David regular sets. In preparation
Garcìa-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116. Amsterdam (1985)
Hofmann, S.:, Local Tb Theorems and applications in PDE. Proceedings of the ICM Madrid, vol. II, pp. 1375–1392. European Math. Soc. (2006)
Hofmann, S.: A local Tb theorem for square functions. Proc. Symp. Pure Math., vol. 79, pp. 175–185. Special volume “ Perspectives in Partial Differential Equations, Harmonic Analysis and Applications”, in honor of the 70th birthday of V. Maz’ya (2008)
Hofmann S., Lacey M., McIntosh A.: The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds. Ann. Math. 156, 623–631 (2002)
Hofmann S., Martell J.M.: Lp bounds for Riesz transforms and square roots associated with second order elliptic operators. Publ. Mat. 47(2), 497–515 (2003)
Hofmann, S., Martell, J.M.: Uniform rectifiability and harmonic measure. Preprint
Hofmann, S., Martell, J.M., Uriarte-Tuero, I.: Uniform rectifiability and harmonic measure II: Poisson kernels in Lp imply uniform rectifiability. In preparation
Hofmann, S., McIntosh, A.: The solution of the Kato problem in two dimensions. In: Proceedings of the Conference on Harmonic Analysis and PDE held in El Escorial, Spain in July 2000. Publ. Mat. Vol. extra, pp. 143–160 (2002)
Hytönen, T., Martikainen, H.: Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. Preprint
Hytönen, T., Martikainen, H.: On general local Tb theorems. Preprint
Hytönen T., McIntosh A., Portal P.: Kato’s square root problem in Banach spaces. J. Funct. Anal. 254, 675–726 (2008)
Hytönen T., Weis L.: A T1 theorem for integral transformations with operator-valued kernel. J. Reine Angew. Math. 599, 155–200 (2006)
Hytönen T.: An operator-valued Tb theorem. J. Funct. Anal. 234, 420–463 (2006)
Journé J.-L.: Remarks on Kato’s square root problem Conference on Mathematical Analysis (El Escorial 1989). Publ. Math. 35(1), 299–321 (1991)
John F., Nirenberg L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426 (1961)
Kato T.: Fractional powers of dissipative operators. J. Math. Soc. Japan 13, 246–274 (1961)
Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1966)
Kenig, C.: Harmonic analysis techniques for second order elliptic boundary value problems, vol. 83. In: CBMS Regional Conference Series in Mathematics. AMS, Providence (1994)
Kenig, C.: Featured review: the solution of the Kato square root problem for second order elliptic operators on \({\mathbb{R}^n}\) . Math. Rev. MR1933726 (2004c:47096c) (2004)
McIntosh, A.: Square roots of operators and applications to hyperbolic PDE’s. Miniconference on Operator Theory and PDE. Proc. of the CMA, ANU Canberra, vol. 5, pp. 124–136 (1984)
McIntosh, A.: When are singular integral operators bounded? Miniconference on Geometry and PDEs 1985. Proc. of the CMA, ANU Canberra, vol. 10, pp. 141–149 (1985)
McIntosh, A.: The square root problem for elliptic operators. Functional Analytic Methods for Partial Differential Equation. Lecture Notes in Mathematics, vol. 1450, pp. 122–140. Springer, Berlin (1990)
McIntosh, A., Meyer, Y.: Algébres d’opérateurs définis par des intégrales singuliéres. C. R. Acad. Sci. Paris 301(Série 1), 395–397 (1985)
Mikhlin S.G.: Multidimensional singular integrals and integral equations Translated from the Russian by WJA Whyte. Pergamon Press, Oxford (1965)
Mattila P., Melnikov M., Verdera J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. Math. (2) 144(1), 127–136 (1996)
Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Nazarov F., Treil S., Volberg A.: Accretive system Tb-theorems on nonhomogeneous spaces. Duke Math. J. 113(2), 259–312 (2002)
Nazarov F., Treil S., Volberg A.: The Tb-theorem on non-homogeneous spaces. Acta Math. 190(2), 151–239 (2003)
Peetre J.: On convolution operators leaving Lp, λ spaces invariant. Ann. Mat. Pura Appl. (4) 72, 295–304 (1966)
Semmes S.: Square function estimates and the T (b) Theorem. Proc. Am. Math. Soc. 110(3), 721–726 (1990)
Spanne S.: Sur l’interpolation entre les espaces \(\mathcal {L}_{{k}^{^{p \Phi}}}\). Ann. Scuola Norm. Sup. Pisa (3) 20, 625–648 (1966)
Stein, E.M.: Singular integrals, harmonic functions, and differentiability properties of functions of several variables. Singular integrals. Proc. Sympos. Pure Math., Chicago, 1966, pp. 316–335. Amer. Math. Soc., Providence (1967)
Stein, E.M.: Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1970)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princteon University Press, Princeton (1970)
Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton (1993)
Tolsa X.: Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math. 190(1), 105–149 (2003)
Verchota G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984)
Volberg, A.: Calderón-Zygmund capacities and operators on nonhomogeneous spaces. In: CBMS Regional Conference Series in Mathematics, vol. 100. Published for the Conference Board of the Math. Sciences, Washington, DC. Amer. Math. Soc., Providence (2003)
Acknowledgments
Some of the material on local Tb theory for square functions is taken from the first named author’s ICM lecture [54]. The second author thanks Pierre Portal for helpful contributions to the manuscript, and to Maren Schmalmack for the diagram. Both authors wish to express their appreciation to Pascal Auscher with whom we have maintained a long and fruitful collaboration on topics covered by this survey.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Neil Trudinger.
Steve Hofmann was supported by the National Science Foundation; Alan McIntosh was supported by the Australian Government through the Australian Research Council.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Hofmann, S., McIntosh, A. Boundedness and applications of singular integrals and square functions: a survey. Bull. Math. Sci. 1, 201–244 (2011). https://doi.org/10.1007/s13373-011-0014-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13373-011-0014-3