Square root problem for divergence operators and related topics

Auscher, Pascal ; Tchamitchian, Philippe

Astérisque, no. 249 (1998) , 180 p.

MR Zbl | 11 citations dans Numdam

@book{AST_1998__249__R1_0,
     author = {Auscher, Pascal and Tchamitchian, Philippe},
     title = {Square root problem for divergence operators and related topics},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {249},
     year = {1998},
     mrnumber = {1651262},
     zbl = {0909.35001},
     language = {en},
     url = {http://www.numdam.org/item/AST_1998__249__R1_0/}
}

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%0 Book
%A Auscher, Pascal
%A Tchamitchian, Philippe
%T Square root problem for divergence operators and related topics
%S Astérisque
%D 1998
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Auscher, Pascal; Tchamitchian, Philippe. Square root problem for divergence operators and related topics. Astérisque, no. 249 (1998), 180 p. http://numdam.org/item/AST_1998__249__R1_0/

Bibliographie
Cité par

[1] D. Albrecht, X. Duong and A. Mcintosh - Operator theory and harmonic analysis, Workshop in Analysis and Geometry 1995, Part III, Proc. of the CMA (Australian National Univ., Canberra), December 1996, p. 77-136. | MR | Zbl

[2] G. Alexopoulos - La conjecture de Kato pour les opérateurs différentiels elliptiques à coefficients périodiques, C. R. Acad. Sci. Paris 312, Série 1 (1991), p. 263-266. | MR | Zbl

[3] J. Angeletti, S. Mazet and P. Tchamitchian - Analysis of second order elliptic operators without boundary conditions with $V M O$ or Hölderian coefficients, Multiscale wavelets methods for partial differential equations (Dahmen, Kurdila, Oswald, ed.), Wavelets and its Applications, vol. 6, Acad. Press, 1997. | MR | DOI

[4] D. Aronson - Bounds for fundamental solutions of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), p. 890-896. | MR | Zbl | DOI

[5] P. Auscher - Regularity theorems and heat kernels for elliptic operators, J. London. Math. Soc. 54 (1996), p. 284-296. | MR | Zbl | DOI

[6] P. Auscher, L. Barthélémy, P. Bénilan and E. M. Ouhabaz - Absence de la $L^{\infty}$ contractivité pour les semi-groupes associés aux opérateurs elliptiques complexes sous forme divergence, Prépublication no 97-41, UMR 6623, Besançon. | Zbl

[7] P. Auscher, T. Coulhon and P. Tchamitchian - Absence de principe du maximum pour certaines équations paraboliques complexes, Coll. Math. 171 (1996), p. 87-95. | MR | Zbl | EuDML

[8] P. Auscher, A. Mcintosh and A. Nahmod - Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), p. 375-403. | MR | Zbl | DOI

[9] P. Auscher, A. Mcintosh and A. Nahmod - The square root problem of Kato in one dimension, and first order systems, Indiana Univ. Math. J. 46 (1997), p. 659-695. | MR | Zbl | DOI

[10] P. Auscher, A. Mcintosh and P. Tchamitchian - Noyau de la chaleur d'opérateurs elliptiques complexes, Math. Research Letters 1 (1994), p. 37-45. | MR | Zbl | DOI

[11] P. Auscher, A. Mcintosh and P. Tchamitchian - Heat kernel of complex elliptic operators and applications, J. Fund. Anal. 152 (1998), p. 22-73. | MR | Zbl | DOI

[12] P. Auscher and P. Tchamitchian - Ondelettes et conjecture de Kato, C. R. Acad. Sci. Paris 313, Série 1 (1991), p. 63-66. | MR | Zbl

[13] P. Auscher and P. Tchamitchian - Conjecture de Kato sur les ouverts de $ℝ$ , Rev. Mat. Iberoamericana 8 (1992), p. 149-199. | MR | Zbl | EuDML | DOI

[14] P. Auscher and P. Tchamitchian - Calcul fonctionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux), Ann. Inst. Fourier 45 (1995), p. 721-778. | MR | Zbl | EuDML | Numdam | DOI

[15] P. Auscher and P. Tchamitchian - Square roots of divergence operators, square functions and singular integrals, Math. Research Letters 3 (1996), p. 429-437. | MR | Zbl | DOI

[16] A. Bensoussan, J.-L. Lions and G. Papanicolaou - Asymptotic Analysis for periodic structures, Studies in Math. its Applications, North Holland, 1978. | MR | Zbl

[17] A. Calderón - Commutators of singular integral operators, Proc. Nat. Acad. Sci. USA (1974), p. 1324-1327.

[18] M. Christ and J.-L. Journé - Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), p. 51-80. | MR | Zbl | DOI

[19] R. Coifman, D. Deng and Y. Meyer - Domaine de la racine carrée de certains opérateurs différentiels accrétifs, Ann. Inst. Fourier 33 (1983), p. 123-134. | MR | Zbl | EuDML | Numdam | DOI

[20] R. Coifman, A. Mcintosh and Y. Meyer - L'intégrale de Cauchy définit un opérateur borné sur $L^{2} (ℝ)$ pour les courbes lipschitziennes, Ann. Math. 116 (1982), p. 361-387. | MR | Zbl | DOI

[21] R. Coifman and Y. Meyer - Non-linear harmonic analysis and PDE, Beijing Lectures in Harmonic Analysis (E. M. Stein, ed.), Annals of Math. Studies, vol. 112, Princeton Univ. Press, 1986. | MR | Zbl

[22] R. Coifman and Y. Meyer - Ondelettes et opérateurs, vol. 3, Hermann, 1991. | MR | Zbl

[23] R. Coifman and S. Semmes - Real-analytic operator-valued functions defined in $B M O$ , Volume in honor of M. Cotlar (C. Sadosky, ed.), Marcel Dekker, 1991. | Zbl | MR

[24] T. Coulhon and X. Duong - Riesz transforms for $1 \leq p \leq 2$ , To appear in Trans. Amer. Math. Soc. | MR | Zbl

[25] G. David - Wavelets and singular integrals on curves and surfaces, Lecture Notes in Math, vol. 1565, Springer-Verlag, 1991. | MR | Zbl

[26] G. David and J.-L. Journé - A boundedness criterion for generalized Calderón-Zygmund operators, Ann. Math. 120 (1984), p. 371-398. | MR | Zbl | DOI

[27] G. David, J.-L. Journé and S. Semmes - Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), p. 1-56. | MR | Zbl | EuDML | DOI

[28] E. Davies - Heat kernels and spectral theory, Cambridge Univ. Press, 1992. | MR | Zbl

[29] E. Davies - Uniformly elliptic operators with measurable coefficients, J. Fund. Anal. 132 (1995), p. 141-169. | MR | Zbl | DOI

[30] E. Davies - Limits on $L^{p}$ regularity of selfadjoint elliptic operators, J. Diff. Eqns. 135 (1997), p. 83-102. | Zbl | MR | DOI

[31] E. De Giorgi - Sulla differenziabilità e l'analyticità delle estremali degli integrali multipli regolari, Mem. Acad. Sci. Torino cl. Sci. Fis. Mat. Nat. 3 (1957), p. 25-43. | MR | Zbl

[32] X. Duong and A. Mcintosh - The $L^{p}$ boundedness of Riesz transforms associated with divergence form operators, 1998, preprint, Macquarie University. | Zbl

[33] X. Duong and D. Robinson - Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, p. 89-128. | MR | Zbl | DOI

[34] E. Fabes and D. Stroock - A new proof of Moser's parabolic inequality via the old ideas of Nash, Arch. Rat. Mech. Anal. 96 (1986), p. 327-338. | MR | Zbl | DOI

[35] E. Fabes, D. Jerison and C. Kenig - Multilinear Littlewood-Paley estimates with applications to partial differential equations, Proc. Nat. Acad. Sci. USA 79 (1982), p. 5746-5750. | MR | Zbl | DOI

[36] E. Fabes, D. Jerison and C. Kenig - Necessary and sufficient conditions for the absolute continuity of elliptic-harmonic measure, Ann. Math. 119 (1984), p. 121-141. | MR | Zbl | DOI

[37] E. Fabes, D. Jerison and C. Kenig - Square functions and partial differential equations, Amer. J. of Math. 107 (1985), p. 1325-1367. | Zbl | MR | DOI

[38] M. Giaquinta - Multiple integrals in the calculus of variations and non-linear elliptic systems, Annals of Math. Studies, vol. 105, Princeton Univ. Press, 1983. | MR | Zbl

[39] M. Giaquinta - Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics, Birkhauser, ETH Zürich, 1993. | Zbl | MR

[40] D. Gilbarg and N. Trudinger - Elliptic partial differential equations of second order, Springer Verlag, 1983. | MR | Zbl | DOI

[41] D. Goldberg - A local version of the real Hardy spaces, Duke Math. J. 46 (1979), p. 27-42. | MR | Zbl | DOI

[42] P. Grisvard - Caractérisation de quelques espaces d'interpolation, Arch. Rat. Mech. Anal. 25 (1967), p. 40-63. | MR | Zbl | DOI

[43] M. Grüter and K.-O. Widman - The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), p. 303-342. | MR | Zbl | EuDML | DOI

[44] V. Jikov, S. Kozlov and O. Oleinik - Homogenization of differential operators and integral functionals, Springer Verlag, 1994. | MR | DOI

[45] J.-L. Journé - Remarks on the square root problem, Pub. Math. 35 (1991), p. 299-321. | Zbl | MR | EuDML | DOI

[46] T. Kato - Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan 5 (1953), p. 208-234. | MR | Zbl | DOI

[47] T. Kato - Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), p. 246-274. | MR | Zbl | DOI

[48] T. Kato - Perturbation theory for linear operators, Springer Verlag, New York, 1966. | MR | Zbl

[49] C. Kenig - Harmonic Analysis techniques for second order elliptic boundary value problems, CBMS - conference lecture notes, vol. 83, AMS, Providence RI, 1995. | MR | Zbl

[50] C. Kenig and Y. Meyer - The Cauchy integral on lipschitz curves and the square root of second order accretive operators are the same, Recent progress in Fourier analysis (Peral, ed.), Math. Studies, vol. 111, North Holland, 1985, p. 123-145. | MR | Zbl

[51] C. Kenig and J. Pipher - The Neumann problem for elliptic equations with non-smooth coefficients, Inventiones Matematicae 113 (1993), p. 447-509. | MR | Zbl | EuDML | DOI

[52] J.-L. Lions - Espaces d'interpolation et domaines de puissances fractionnaires, J. Math. Soc. Japan 14 (1962), p. 233-241. | MR | Zbl | DOI

[53] V. Maz'Ya, S. Nazarov and B. Plamenevskii - Absence of De Giorgi-type theorems for strongly elliptic equations with complex coefficients, J. Math. Sov. 28 (1985), p. 726-739. | Zbl | DOI

[54] A. Mcintosh - On the comparability of $A^{1 / 2}$ and $A^{* 1 / 2}$ , Proc. Amer. Math. Soc. 32 (1972), p. 430-434. | MR | Zbl | DOI

[55] A. Mcintosh - On representing closed accretive sesquilinear forms as $(A^{1 / 2} u, A^{* 1 / 2} v)$ , Collège de France Seminar, Volume III (H. Brezis and J.-L. Lions, ed.), Research Notes in Mathematics, no. 70, Pitman, 1982, p. 252-267. | MR | Zbl

[56] A. Mcintosh - Square roots of operators and applications to hyperbolic PDE's, Miniconference on operator theory and partial differential equations (Canberra), Center for Math. and Appl., Australian National Univ., 1983. | MR | Zbl

[57] A. Mcintosh - Square roots of elliptic operators, J. Fund. Anal. 61 (1985), p. 307-327. | MR | Zbl | DOI

[58] A. Mcintosh - Operators which have an $H^{\infty}$ functional calculus, Miniconference on operator theory and partial differential equations (Canberra), Center for Math. and Appl., vol. 14, Australian National Univ., 1986, p. 210-231. | MR | Zbl

[59] A. Mcintosh - Clifford algebras and the higher dimensional Cauchy integral, Approximation and function spaces, Banach Center publications, vol. 22, Polish Scientific Publishers, 1989. | MR | Zbl

[60] A. Mcintosh - The square root problem for elliptic operators, Functional analytic methods for partial differential equations (Berlin), Lect. Notes in Math., vol. 1450, Springer-Verlag, 1990, p. 122-140. | MR | Zbl | DOI

[61] A. Mcintosh and Y. Meyer - Algèbres d'opérateurs définis par des intégrales singulières, C. R. Acad. Sci. Paris 301, Série 1 (1985), p. 395-397. | MR | Zbl

[62] A. Mcintosh and A. Yagi - Operators of type $ω$ without a bounded $H^{\infty}$ functional calculus, Miniconference on Operators in Analysis (Canberra), Center for Math. and Appl., vol. 24, Australian National Univ., 1989, p. 159-174. | MR | Zbl

[63] Y. Meyer - Ondelettes et opérateurs, Hermann, volumes 1 & 2, 1990. | Zbl

[64] C. Morrey - Multiple integrals in the calculus of variations, Springer Verlag, 1966. | MR | Zbl

[65] J. Moser - On Harnack's theorem for elliptic partial differential equations, Comm. Pure & Appli. Math. 14 (1961), p. 577-691. | MR | Zbl | DOI

[66] J. Nash - Continuity of solutions of parabolic and elliptic equations, Amer. J. of Math. 80 (1958), p. 931-954. | MR | Zbl | DOI

[67] F. Porper and S. Eidel'Man - Two sided estimates of fundamental solutions of second order parabolic equations and some applications, Russian Math. Surveys 39 (1984), no. 3, p. 119-178. | Zbl | DOI

[68] D. Robinson - Elliptic operators and Lie groups, Oxford Mathematical Monographs (1991). | MR | Zbl

[69] S. Semmes - Square function estimates and the $T (b)$ Theorem, Proc. Amer. Math. Soc. 110 (1990), no. 3, p. 721-726. | MR | Zbl

[70] I. Sneiberg - Spectral properties of linear operators in interpolation families of Banach spaces, mat. Issled. 9 (1974), p. 214-229. | MR | Zbl

[71] E. Stein - Singular integrals and differentiability of functions, Princeton Univ. Press, 1970. | MR | Zbl

[72] E. Stein - Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Math. Studies, vol. 63, Princeton Univ. Press, 1970. | MR | Zbl

[73] E. Stein - The development of square function estimates in the work of Antoni Zygmund, Bull. Amer. Math. Soc. 7 (1982), p. 359-376. | MR | Zbl | DOI

[74] E. Stein - Harmonic analysis, Princeton Univ. Press, 1993. | MR | Zbl

[75] E. Stein and G. Weiss - Introduction to Fourier analysis on Euclidean space, Princeton Univ. Press, 1971. | MR

[76] R. Strichartz - Multipliers on fractional Sobolev spaces, J. Math. Mech 16 (1967), p. 1031-1060. | MR | Zbl

[77] H. Tanabe - Equations of evolution, Pitman, London, 1979. | MR | Zbl

[78] M. Taylor - Analysis on Morrey spaces and applications to Navier-Stokes equations and other evolution equations, Comm. Partial & Diff. Equa. 17 (1992), p. 1407-1456. | MR | Zbl | DOI

[79] H. Triebel - Theory of function spaces, Monographs in Math., vol. 78, Birkhauser, 1983. | MR | Zbl

[80] N. Varopoulos, L. Saloff-Coste and T. Coulhon - Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, 1993. | Zbl

[81] A. Yagi - Coïncidence entre des espaces d'interpolation et des domaines de puissance fractionnaires d'opérateurs, C. R. Acad. Sci. Paris 299, Série 1 (1984), p. 173-176. | Zbl

[82] Zygmund - Trigonometric series, second ed., Cambridge Univ. Press, 1959. | Zbl