Tome (1998) no. 249

Square root problem for divergence operators and related topics

Auscher, Pascal ; Tchamitchian, Philippe

Astérisque, no. 249 (1998), 180 p.

@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},
     zbl = {0909.35001},
     mrnumber = {1651262},
     language = {en},
     url = {http://www.numdam.org/item/AST_1998__249__R1_0}
}

Auscher, Pascal; Tchamitchian, Philippe. Square root problem for divergence operators and related topics. Astérisque, no. 249 (1998), 180 p. http://www.numdam.org/item/AST_1998__249__R1_0/

Bibliographie

[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 1394696 | Zbl 0903.47010

[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 1089711 | Zbl 0765.35014

[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. | Article | MR 1475008

[4] D. Aronson - Bounds for fundamental solutions of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), p. 890-896. | Article | MR 217444 | Zbl 0153.42002

[5] P. Auscher - Regularity theorems and heat kernels for elliptic operators, J. London. Math. Soc. 54 (1996), p. 284-296. | Article | MR 1405056 | Zbl 0863.35020

[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 0951.31006

[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 1397370 | Zbl 0960.35011

[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. | Article | MR 1481596 | Zbl 0903.47011

[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. | Article | MR 1488331 | Zbl 0911.47041

[10] P. Auscher, A. Mcintosh and P. Tchamitchian - Noyau de la chaleur d'opérateurs elliptiques complexes, Math. Research Letters 1 (1994), p. 37-45. | Article | MR 1258487 | Zbl 0827.35033

[11] P. Auscher, A. Mcintosh and P. Tchamitchian - Heat kernel of complex elliptic operators and applications, J. Fund. Anal. 152 (1998), p. 22-73. | Article | MR 1600066 | Zbl 0919.35035

[12] P. Auscher and P. Tchamitchian - Ondelettes et conjecture de Kato, C. R. Acad. Sci. Paris 313, Série 1 (1991), p. 63-66. | MR 1119909 | Zbl 0733.47034

[13] P. Auscher and P. Tchamitchian - Conjecture de Kato sur les ouverts de $ℝ$ , Rev. Mat. Iberoamericana 8 (1992), p. 149-199. | Article | MR 1191343 | Zbl 0794.35035

[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. | Article | Numdam | MR 1340951 | Zbl 0819.35028

[15] P. Auscher and P. Tchamitchian - Square roots of divergence operators, square functions and singular integrals, Math. Research Letters 3 (1996), p. 429-437. | Article | MR 1397689 | Zbl 0858.47019

[16] A. Bensoussan, J.-L. Lions and G. Papanicolaou - Asymptotic Analysis for periodic structures, Studies in Math. its Applications, North Holland, 1978. | MR 503330 | Zbl 0404.35001

[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. | Article | MR 906525 | Zbl 0645.42017

[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. | Article | Numdam | MR 699490 | Zbl 0497.35088

[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. | Article | MR 672839 | Zbl 0497.42012

[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 864370 | Zbl 0623.47052

[22] R. Coifman and Y. Meyer - Ondelettes et opérateurs, vol. 3, Hermann, 1991. | MR 1160989 | Zbl 0745.42012

[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. | MR 1044783 | Zbl 0709.47012

[24] T. Coulhon and X. Duong - Riesz transforms for $1 \leq p \leq 2$ , To appear in Trans. Amer. Math. Soc. | MR 1458299 | Zbl 0973.58018

[25] G. David - Wavelets and singular integrals on curves and surfaces, Lecture Notes in Math, vol. 1565, Springer-Verlag, 1991. | MR 1123480 | Zbl 0764.42019

[26] G. David and J.-L. Journé - A boundedness criterion for generalized Calderón-Zygmund operators, Ann. Math. 120 (1984), p. 371-398. | Article | MR 763911 | Zbl 0567.47025

[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. | Article | MR 850408 | Zbl 0604.42014

[28] E. Davies - Heat kernels and spectral theory, Cambridge Univ. Press, 1992. | MR 990239 | Zbl 0699.35006

[29] E. Davies - Uniformly elliptic operators with measurable coefficients, J. Fund. Anal. 132 (1995), p. 141-169. | Article | MR 1346221 | Zbl 0839.35034

[30] E. Davies - Limits on $L^{p}$ regularity of selfadjoint elliptic operators, J. Diff. Eqns. 135 (1997), p. 83-102. | Article | MR 1434916 | Zbl 0871.35020

[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 93649 | Zbl 0084.31901

[32] X. Duong and A. Mcintosh - The $L^{p}$ boundedness of Riesz transforms associated with divergence form operators, 1998, preprint, Macquarie University. | Zbl 1193.42089

[33] X. Duong and D. Robinson - Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, p. 89-128. | Article | MR 1419418 | Zbl 0932.47013

[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. | Article | MR 855753 | Zbl 0652.35052

[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. | Article | MR 674919 | Zbl 0501.35014

[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. | Article | MR 736563 | Zbl 0551.35024

[37] E. Fabes, D. Jerison and C. Kenig - Square functions and partial differential equations, Amer. J. of Math. 107 (1985), p. 1325-1367. | Article | MR 815765 | Zbl 0655.35007

[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 717034 | Zbl 0516.49003

[39] M. Giaquinta - Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics, Birkhauser, ETH Zürich, 1993. | MR 1239172 | Zbl 0786.35001

[40] D. Gilbarg and N. Trudinger - Elliptic partial differential equations of second order, Springer Verlag, 1983. | Article | MR 737190 | Zbl 0562.35001

[41] D. Goldberg - A local version of the real Hardy spaces, Duke Math. J. 46 (1979), p. 27-42. | Article | MR 523600 | Zbl 0409.46060

[42] P. Grisvard - Caractérisation de quelques espaces d'interpolation, Arch. Rat. Mech. Anal. 25 (1967), p. 40-63. | Article | MR 213864 | Zbl 0187.05901

[43] M. Grüter and K.-O. Widman - The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), p. 303-342. | Article | MR 657523 | Zbl 0485.35031

[44] V. Jikov, S. Kozlov and O. Oleinik - Homogenization of differential operators and integral functionals, Springer Verlag, 1994. | Article | MR 1329546

[45] J.-L. Journé - Remarks on the square root problem, Pub. Math. 35 (1991), p. 299-321. | Article | MR 1103623 | Zbl 0739.47009

[46] T. Kato - Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan 5 (1953), p. 208-234. | Article | MR 58861 | Zbl 0052.12601

[47] T. Kato - Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), p. 246-274. | Article | MR 138005 | Zbl 0113.10005

[48] T. Kato - Perturbation theory for linear operators, Springer Verlag, New York, 1966. | MR 203473 | Zbl 0435.47001 | Zbl 0148.12601

[49] C. Kenig - Harmonic Analysis techniques for second order elliptic boundary value problems, CBMS - conference lecture notes, vol. 83, AMS, Providence RI, 1995. | MR 1282720 | Zbl 0812.35001

[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 848144 | Zbl 0641.47039

[51] C. Kenig and J. Pipher - The Neumann problem for elliptic equations with non-smooth coefficients, Inventiones Matematicae 113 (1993), p. 447-509. | Article | MR 1231834 | Zbl 0807.35030

[52] J.-L. Lions - Espaces d'interpolation et domaines de puissances fractionnaires, J. Math. Soc. Japan 14 (1962), p. 233-241. | Article | MR 152878 | Zbl 0108.11202

[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. | Article | Zbl 0562.35030

[54] A. Mcintosh - On the comparability of $A^{1 / 2}$ and $A^{* 1 / 2}$ , Proc. Amer. Math. Soc. 32 (1972), p. 430-434. | Article | MR 290169 | Zbl 0248.47020

[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 670278 | Zbl 0515.47013

[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 757577 | Zbl 0565.35064

[57] A. Mcintosh - Square roots of elliptic operators, J. Fund. Anal. 61 (1985), p. 307-327. | Article | MR 820618 | Zbl 0592.47043

[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 912940 | Zbl 0634.47016

[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 1097197 | Zbl 0689.42017

[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. | Article | MR 1084605 | Zbl 0723.47032

[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 808636 | Zbl 0584.47030

[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 1060121 | Zbl 0709.47016

[63] Y. Meyer - Ondelettes et opérateurs, Hermann, volumes 1 & 2, 1990. | Zbl 0694.41037

[64] C. Morrey - Multiple integrals in the calculus of variations, Springer Verlag, 1966. | MR 202511 | Zbl 1213.49002 | Zbl 0142.38701

[65] J. Moser - On Harnack's theorem for elliptic partial differential equations, Comm. Pure & Appli. Math. 14 (1961), p. 577-691. | Article | MR 159138 | Zbl 0111.09302

[66] J. Nash - Continuity of solutions of parabolic and elliptic equations, Amer. J. of Math. 80 (1958), p. 931-954. | Article | MR 100158 | Zbl 0096.06902

[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. | Article | Zbl 0582.35052

[68] D. Robinson - Elliptic operators and Lie groups, Oxford Mathematical Monographs (1991). | MR 1144020 | Zbl 0747.47030

[69] S. Semmes - Square function estimates and the $T (b)$ Theorem, Proc. Amer. Math. Soc. 110 (1990), no. 3, p. 721-726. | MR 1028049 | Zbl 0719.42023

[70] I. Sneiberg - Spectral properties of linear operators in interpolation families of Banach spaces, mat. Issled. 9 (1974), p. 214-229. | MR 634681 | Zbl 0314.46033

[71] E. Stein - Singular integrals and differentiability of functions, Princeton Univ. Press, 1970. | MR 290095 | Zbl 0207.13501

[72] E. Stein - Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Math. Studies, vol. 63, Princeton Univ. Press, 1970. | MR 252961 | Zbl 0193.10502

[73] E. Stein - The development of square function estimates in the work of Antoni Zygmund, Bull. Amer. Math. Soc. 7 (1982), p. 359-376. | Article | MR 663787 | Zbl 0526.01021

[74] E. Stein - Harmonic analysis, Princeton Univ. Press, 1993. | MR 1232192 | Zbl 0821.42001

[75] E. Stein and G. Weiss - Introduction to Fourier analysis on Euclidean space, Princeton Univ. Press, 1971. | MR 304972

[76] R. Strichartz - Multipliers on fractional Sobolev spaces, J. Math. Mech 16 (1967), p. 1031-1060. | MR 215084 | Zbl 0145.38301

[77] H. Tanabe - Equations of evolution, Pitman, London, 1979. | MR 533824 | Zbl 0417.35003

[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. | Article | MR 1187618 | Zbl 0771.35047

[79] H. Triebel - Theory of function spaces, Monographs in Math., vol. 78, Birkhauser, 1983. | MR 781540 | Zbl 0546.46028 | Zbl 0546.46027

[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 1179.22009

[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 0563.46042

[82] Zygmund - Trigonometric series, second ed., Cambridge Univ. Press, 1959. | Zbl 0085.05601