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Sobolev Inequalities in Familiar and Unfamiliar Settings

  • Laurent Saloff-Coste
Part of the International Mathematical Series book series (IMAT, volume 8)

Abstract

The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts.

Keywords

Heat Kernel Cayley Graph Sobolev Inequality Dirichlet Form Volume Growth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alexopoulos, G., Lohoué, N.: Sobolev inequalities and harmonic functions of polyno mial growth. J. London Math. Soc. (2) 48, no. 3, 452–464 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aronson, D.G., Serrin, J.: Local behavior of solutions of quasilinear parabolic equa tions. Arch. Ration. Mech. Anal. 25, 81–122 (1967)MATHMathSciNetGoogle Scholar
  3. 3.
    Aronson, D.G., Serrin, J.: A maximum principle for nonlinear parabolic equations. Ann. Sc. Norm. Super. Pisa (3) 21, 291–305 (1967)MATHMathSciNetGoogle Scholar
  4. 4.
    Bakry, D., Coulhon, T., Ledoux, M., Saloff-Coste, L.: Sobolev inequalities in disguise. Indiana Univ. Math. J. 44, no. 4, 1033–1074 (1995)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bakry, D., Ledoux, M.: Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator. Duke Math. J. 85, no. 1, 253–270 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Barlow, M., Coulhon, T., Grigor'yan, A.: Manifolds and graphs with slow heat kernel decay. Invent. Math. 144, no. 3, 609–649 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Barlow, M.T., Coulhon, T., Kumagai, T.: Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Commun. Pure Appl. Math. 58, no. 12, 1642–1677 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bendikov, A., Coulhon, T., Saloff-Coste, L.: Ultracontractivity and embedding into L . Math. Ann. 337, no. 4, 817–853 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bendikov, A., Maheux, P.: Nash type inequalities for fractional powers of nonnegative self-adjoint operators. Trans. Am. Math. Soc. 359, no. 7, 3085–3097 (electronic) (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Biroli, M., Mosco, U.: A Saint-Venant type principle for Dirichlet forms on discon tinuous media. Ann. Mat. Pura Appl. (4) 169, 125–181 (1995)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Biroli, M., Mosco, U.: Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6, no. 1, 37–44 (1995)MATHMathSciNetGoogle Scholar
  12. 12.
    Biroli, M., Mosco, U.: Sobolev inequalities on homogeneous spaces. Potential Anal. 4, no. 4, 311–324 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bougerol, P., Élie, L.: Existence of positive harmonic functions on groups and on covering manifolds. Ann. Inst. H. Poincaré Probab. Statist. 31, no. 1, 59–80 (1995)MATHGoogle Scholar
  14. 14.
    Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15, no. 2, 213–230 (1982)MATHMathSciNetGoogle Scholar
  15. 15.
    Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov tran sition functions. Ann. Inst. H. Poincaré Probab. Statist. 23, no. 2, suppl., 245–287 (1987)MathSciNetGoogle Scholar
  16. 16.
    Carron, G.: Formes harmoniques L 2 sur les variétés non-compactes. Rend. Mat. Appl. (7) 21, no. 1–4, 87–119 (2001)MATHMathSciNetGoogle Scholar
  17. 17.
    Carron, G.: Inégalités isopérimétriques de Faber-Krahn et conséquences. In: Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) Sémin. Congr. Vol. 1, pp. 205–232. Soc. Math. France, Paris (1996)Google Scholar
  18. 18.
    Carron, G.: Une suite exacte en L 2-cohomologie. Duke Math. J. 95, no. 2, 343–372 (1998)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Carron, G.: L 2-cohomologie et inégalités de Sobolev. Math. Ann. 314, no. 4, 613–639 (1999)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Chavel, I.: Eigenvalues in Riemannian geometry Academic Press Inc., Orlando, FL (1984)Google Scholar
  21. 21.
    Chavel, I.: Riemannian Geometry-a Modern Introduction. Cambridge Univ. Press, Cambridge (1993)MATHGoogle Scholar
  22. 22.
    Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, no. 1, 15–53 (1982)MATHMathSciNetGoogle Scholar
  23. 23.
    Cheng, S.Y., Li, P., Yau, Sh.T.: On the upper estimate of the heat kernel of a complete Riemannian manifold. Am. J. Math. 103, no. 5, 1021–1063 (1981)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Colding, T.H., Minicozzi, W.P.II: Harmonic functions on manifolds. Ann. Math. (2) 146, no. 3, 725–747 (1997)MATHMathSciNetGoogle Scholar
  25. 25.
    Colding, T.H., Minicozzi, W.P.II: Weyl type bounds for harmonic functions. Invent. Math. 131, no. 2, 257–298 (1998)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Coulhon, T.: Dimension à l'infini d'un semi-groupe analytique. Bull. Sci. Math. 114, no. 4, 485–500 (1990)MATHMathSciNetGoogle Scholar
  27. 27.
    Coulhon, T.: Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141, no. 2, 510–539 (1996)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Coulhon, T.: Analysis on infinite graphs with regular volume growth. In: Random Walks and Discrete Potential Theory (Cortona, 1997) Sympos. Math. Vol. XXXIX, pp. 165–187. Cambridge Univ. Press, Cambridge (1999)Google Scholar
  29. 29.
    Coulhon, T., Grigor'yan, A.: Pointwise estimates for transition probabilities of ran dom walks on infinite graphs. In: Fractals in Graz 2001, pp. 119–134. Birkhäuser, Basel (2003)Google Scholar
  30. 30.
    Coulhon, T., Grigor'yan, A., Zucca, F.: The discrete integral maximum principle and its applications. Tohoku Math. J. (2) 57, no. 4, 559–587 (2005)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Coulhon, T., Saloff-Coste, L.: Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoam. 9, no. 2, 293–314 (1993)MATHMathSciNetGoogle Scholar
  32. 32.
    Coulhon, T., Saloff-Coste, L.: Variétés riemanniennes isométriques à l'infini. Rev. Mat. Iberoam. 11, no. 3, 687–726 (1995)MATHMathSciNetGoogle Scholar
  33. 33.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge (1989)MATHGoogle Scholar
  34. 34.
    DeGiorgi, E.: Sulla differenziabilità e l'analiticità delle estremali degli integrali mul tipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3, 25–43 (1957)MathSciNetGoogle Scholar
  35. 35.
    de la Harpe, P.: Topics in Geometric Group Theory. University of Chicago Press, Chicago, IL (2000)MATHGoogle Scholar
  36. 36.
    Deny, J.: Méthodes hilbertiennes en théorie du potentiel. In: Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969), pp. 121–201. Edizioni Cremonese, Rome (1970)Google Scholar
  37. 37.
    Duong, T.X., Ouhabaz, ElM., Sikora, A.: Plancherel type estimates and sharp spectral multipliers. J. Funct. Anal. 196, no. 2, 443–485 (2002)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Duong, T.X., Ouhabaz, ElM., Sikora, A.: Spectral multipliers for self-adjoint opera tors. In: Geometric Analysis and Applications (Canberra (2000)),pp. 56–66. Austral. Nat. Univ., Canberra (2001)Google Scholar
  39. 39.
    Franchi, B., Gutiérrez, C.E., Wheeden, R.L.: Weighted Sobolev–Poincaré inequalities for Grushin type operators. Commun. Partial Differ. Equ. 19, no. 3-4, 523–604 (1994)MATHCrossRefGoogle Scholar
  40. 40.
    Fukushima, M., Ōshima, Yō., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (1994)MATHGoogle Scholar
  41. 41.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Springer-Verlag, Berlin (1990)MATHGoogle Scholar
  42. 42.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001)MATHGoogle Scholar
  43. 43.
    Grigor'yan, A.: The heat equation on noncompact Riemannian manifolds (Russian). Mat. Sb. 182, no. 1, 55–87 (1991); English transl.: Math. USSR, Sb. 72, no. 1, 47–77 (1992)MATHGoogle Scholar
  44. 44.
    Grigor'yan, A.: Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10, no. 2, 395–452 (1994)MATHMathSciNetGoogle Scholar
  45. 45.
    Grigor'yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.) 36, no. 2, 135–249 (1999)MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Grigor'yan, A.: Estimates of heat kernels on Riemannian manifolds. In: Spectral Theory and Geometry (Edinburgh (1998)), pp. 140–225. Cambridge Univ. Press, Cambridge (1999)Google Scholar
  47. 47.
    Grigor'yan, A.: Heat kernels on manifolds, graphs and fractals. In: European Congress of Mathematics, Vol. I (Barcelona (2000)), pp. 393–406. Birkhäuser, Basel (2001)Google Scholar
  48. 48.
    Grigor'yan, A.: Heat kernels on weighted manifolds and applications. In: The Ubiq uitous Heat Kernel, pp. 93–191. Am. Math. Soc., Providence, RI (2006)Google Scholar
  49. 49.
    Grigor'yan, A., Saloff-Coste, L.: Heat kernel on connected sums of Riemannian man ifolds. Math. Res. Lett. 6, no. 3-4, 307–321 (1999)MATHMathSciNetGoogle Scholar
  50. 50.
    Grigor'yan, A., Saloff-Coste, L.: Stability results for Harnack inequalities. Ann. Inst. Fourier (Grenoble) 55, no. 3, 825–890 (2005)MATHMathSciNetGoogle Scholar
  51. 51.
    Grigor'yan, A., Saloff-Coste, L.: Heat kernel on manifolds with ends. (2007)Google Scholar
  52. 52.
    Grigor'yan, A., Telcs. A.: Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109, no. 3, 451–510 (2001)MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Grigor'yan, A., Telcs. A.: Harnack inequalities and sub-Gaussian estimates for ran dom walks. Math. Ann. 324, no. 3, 521–556 (2002)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Guivarc'h, Y.: Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101, 333–379 (1973)MATHMathSciNetGoogle Scholar
  56. 56.
    Gutiérrez, C.E., Wheeden, R.: Mean value and Harnack inequalities for degenerate parabolic equations. Colloq. Math. 60/61, no. 1, 157–194 (1990)Google Scholar
  57. 57.
    Gutiérrez, C.E., Wheeden, R.: Harnack's inequality for degenerate parabolic equa tions. Commun. Partial Differ. Equ. 16,no. 4–5, 745–770 (1991)MATHCrossRefGoogle Scholar
  58. 58.
    Gutiérrez, C.E., Wheeden, R.: Bounds for the fundamental solution of degenerate parabolic equations. Commun. Partial Differ. Equ. 17, no. 7–8, 1287–1307 (1992)MATHCrossRefGoogle Scholar
  59. 59.
    Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. no. 688 (2000)Google Scholar
  60. 60.
    Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lect. Notes 5. Am. Math. Soc, Providence, RI (1999)Google Scholar
  61. 61.
    Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51,no. 5, 1437–1481 (2001)MATHMathSciNetGoogle Scholar
  62. 62.
    Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York (2001)Google Scholar
  63. 63.
    Heinonen, J.: Nonsmooth calculus. Bull. Am. Math. Soc. (N.S.) 44, no. 2, 163–232 (electronic), (2007)MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Kanai, M.: Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan 37, no. 3, 391–413 (1985)MATHMathSciNetGoogle Scholar
  65. 65.
    Kanai, M.: Analytic inequalities, and rough isometries between noncompact Rie mannian manifolds. In: Curvature and Topology of Riemannian Manifolds (Katata, (1985), pp. 122–137. Springer, Berlin (1986)CrossRefGoogle Scholar
  66. 66.
    Kanai, M.: Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Japan 38, no. 2, 227–238 (1986)MATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Kleiner, B.: A new proof of Gromov's theorem on groups of polynomial growth. (2007) arXiv:0710.4593v4Google Scholar
  68. 68.
    Kuchment, P., Pinchover, Y.: Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds. Trans. Am. Math. Soc. 359, no. 12, 5777– 5815 (electronic) (2007)MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type Am. Math. Soc, Providence, RI (1968)Google Scholar
  70. 70.
    Levin, D., Solomyak, M.: The Rozenblum-Lieb-Cwikel inequality for Markov gener ators. J. Anal. Math. 71, 173–193 (1997)MATHMathSciNetGoogle Scholar
  71. 71.
    Li, P.: Harmonic functions on complete Riemannian manifolds. In: Tsing Hua Lectures on Geometry and Analysis (Hsinchu, (1990–1991), pp. 265–268. Int. Press, Cambridge, MA (1997)Google Scholar
  72. 72.
    Li, P.: Harmonic sections of polynomial growth. Math. Res. Lett. 4, no. 1, 35–44 (1997)MATHMathSciNetGoogle Scholar
  73. 73.
    Li, P., Yau, Sh.T.: On the Schrödinger equation and the eigenvalue problem. Com mun. Math. Phys. 88, no. 3, 309–318 (1983)MATHCrossRefMathSciNetGoogle Scholar
  74. 74.
    Li, P., Yau, Sh.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156,no. 3–4, 153–201 (1986)CrossRefMathSciNetGoogle Scholar
  75. 75.
    Maz'ya, V.G.: Sobolev Spaces. Springer-Verlag, Berlin-Tokyo (1985)Google Scholar
  76. 76.
    Morrey, Ch.B., Jr.: Multiple Integrals in the Calculus of Variations. Springer-Verlag, New York (1966)MATHGoogle Scholar
  77. 77.
    Moser, J.: On Harnack's theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)MATHCrossRefGoogle Scholar
  78. 78.
    Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)MATHCrossRefGoogle Scholar
  79. 79.
    Moser, J.: Correction to: “A Harnack inequality for parabolic differential equations.” Commun. Pure Appl. Math. 20, 231–236 (1967)MATHCrossRefGoogle Scholar
  80. 80.
    Moser, J.: On a pointwise estimate for parabolic differential equations. Commun. Pure Appl. Math. 24, 727–740 (1971)MATHCrossRefGoogle Scholar
  81. 81.
    Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)MATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    Pivarski, M., Saloff-Coste, L.: Small time heat kernel behavior on riemannian com plexeGoogle Scholar
  83. 83.
    Porper, F.O., Eidel'man, S.D.: Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications (Russian). Uspekhi Mat. Nauk 39, no. 3, 107–156 (1984); English transl.: Russ. Math. Surv. 39, no. 3, 119– 178 (1984)MathSciNetGoogle Scholar
  84. 84.
    Rozenblum, G., Solomyak, M.: The Cwikel-Lieb-Rozenblum estimates for generators of positive semigroups and semigroups dominated by positive semigroups (Russian). Algebra Anal. 9, no. 6, 214–236 (1997); English transl.: St. Petersbg. Math. J. 9, no. 6, 1195–1211 (1998)Google Scholar
  85. 85.
    Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. 2, 27–38 (1992)CrossRefGoogle Scholar
  86. 86.
    Saloff-Coste, L.: Parabolic Harnack inequality for divergence form second order dif ferential operators. Potential Anal.4, no. 4, 429–467 (1995)MATHCrossRefMathSciNetGoogle Scholar
  87. 87.
    Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. Cambridge Univ. Press, Cam bridge (2002)MATHGoogle Scholar
  88. 88.
    Saloff-Coste, L.: Analysis on Riemannian co-compact covers. In: Surveys in Differen tial Geometry. Vol. 9, pp. 351–384. Int. Press, Somerville, MA (2004)Google Scholar
  89. 89.
    Semmes, S.: Some Novel Types of Fractal Geometry. Clarendon Press, Oxford (2001)MATHGoogle Scholar
  90. 90.
    Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math.111, 247–302 (1964)MATHCrossRefMathSciNetGoogle Scholar
  91. 91.
    Sikora, A.: Sharp pointwise estimates on heat kernels. Q. J. Math., Oxf. II. Ser.47, no. 187, 371–382 (1996)MATHCrossRefMathSciNetGoogle Scholar
  92. 92.
    Sikora, A.: Riesz transform, Gaussian bounds and the method of wave equation. Math. Z.247, no. 3, 643–662 (2004)MATHCrossRefMathSciNetGoogle Scholar
  93. 93.
    Sobolev, S.L.: On a theorem of functional analysis (Russian). Mat. Sb.46, 471–497 (1938); English transl.: Am. Math. Soc., Transl., II. Ser.34, 39–68 (1963)Google Scholar
  94. 94.
    Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9)75, no. 3, 273–297 (1996)MATHMathSciNetGoogle Scholar
  95. 95.
    Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness andL p-Liouville properties. J. Reine Angew. Math.456, 173–196 (1994)MATHMathSciNetGoogle Scholar
  96. 96.
    Sturm, K.T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math.32, no. 2, 275–312 (1995)MATHMathSciNetGoogle Scholar
  97. 97.
    Sturm, K.T.: The geometric aspect of Dirichlet forms. In: New Directions in Dirichlet Forms, pp. 233–277. Am. Math. Soc., Providence, RI (1998)Google Scholar
  98. 98.
    Telcs, A.: The Art of Random Walks. Springer-Verlag, Berlin (2006)MATHGoogle Scholar
  99. 99.
    van den Dries, L., Wilkie, A.J.: Gromov's theorem on groups of polynomial growth and elementary logic. J. Algebra89, no. 2, 349–374 (1984)MATHCrossRefMathSciNetGoogle Scholar
  100. 100.
    Varopoulos, N.Th.: Hardy-Littlewood theory for semigroups. J. Funct. Anal.63, no. 2, 240–260 (1985)MATHCrossRefMathSciNetGoogle Scholar
  101. 101.
    Varopoulos, N.Th.: Isoperimetric inequalities and Markov chains. J. Funct. Anal.63, no. 2, 215–239 (1985)MATHCrossRefMathSciNetGoogle Scholar
  102. 102.
    Varopoulos, N.Th.: Convolution powers on locally compact groups. Bull. Sci. Math. (2)111, no. 4, 333–342 (1987)MATHMathSciNetGoogle Scholar
  103. 103.
    Varopoulos, N.Th., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Univ. Press, Cambridge (1992)Google Scholar
  104. 104.
    Varopoulos, N.Th.: Théorie du potentiel sur des groupes et des variétés. C. R. Acad. Sci., Paris Sér. I Math.302, no. 6, 203–205 (1986)MATHMathSciNetGoogle Scholar
  105. 105.
    Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Univ. Press, Cambridge (2000)MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Laurent Saloff-Coste
    • 1
  1. 1.Cornell University, Mallot HallIthacaUSA

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