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.
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References
Alexopoulos, G., Lohoué, N.: Sobolev inequalities and harmonic functions of polyno mial growth. J. London Math. Soc. (2) 48, no. 3, 452–464 (1993)
Aronson, D.G., Serrin, J.: Local behavior of solutions of quasilinear parabolic equa tions. Arch. Ration. Mech. Anal. 25, 81–122 (1967)
Aronson, D.G., Serrin, J.: A maximum principle for nonlinear parabolic equations. Ann. Sc. Norm. Super. Pisa (3) 21, 291–305 (1967)
Bakry, D., Coulhon, T., Ledoux, M., Saloff-Coste, L.: Sobolev inequalities in disguise. Indiana Univ. Math. J. 44, no. 4, 1033–1074 (1995)
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)
Barlow, M., Coulhon, T., Grigor'yan, A.: Manifolds and graphs with slow heat kernel decay. Invent. Math. 144, no. 3, 609–649 (2001)
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)
Bendikov, A., Coulhon, T., Saloff-Coste, L.: Ultracontractivity and embedding into L ∞. Math. Ann. 337, no. 4, 817–853 (2007)
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)
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)
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)
Biroli, M., Mosco, U.: Sobolev inequalities on homogeneous spaces. Potential Anal. 4, no. 4, 311–324 (1995)
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)
Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15, no. 2, 213–230 (1982)
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)
Carron, G.: Formes harmoniques L 2 sur les variétés non-compactes. Rend. Mat. Appl. (7) 21, no. 1–4, 87–119 (2001)
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)
Carron, G.: Une suite exacte en L 2-cohomologie. Duke Math. J. 95, no. 2, 343–372 (1998)
Carron, G.: L 2-cohomologie et inégalités de Sobolev. Math. Ann. 314, no. 4, 613–639 (1999)
Chavel, I.: Eigenvalues in Riemannian geometry Academic Press Inc., Orlando, FL (1984)
Chavel, I.: Riemannian Geometry-a Modern Introduction. Cambridge Univ. Press, Cambridge (1993)
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)
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)
Colding, T.H., Minicozzi, W.P.II: Harmonic functions on manifolds. Ann. Math. (2) 146, no. 3, 725–747 (1997)
Colding, T.H., Minicozzi, W.P.II: Weyl type bounds for harmonic functions. Invent. Math. 131, no. 2, 257–298 (1998)
Coulhon, T.: Dimension à l'infini d'un semi-groupe analytique. Bull. Sci. Math. 114, no. 4, 485–500 (1990)
Coulhon, T.: Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141, no. 2, 510–539 (1996)
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)
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)
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)
Coulhon, T., Saloff-Coste, L.: Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoam. 9, no. 2, 293–314 (1993)
Coulhon, T., Saloff-Coste, L.: Variétés riemanniennes isométriques à l'infini. Rev. Mat. Iberoam. 11, no. 3, 687–726 (1995)
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge (1989)
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)
de la Harpe, P.: Topics in Geometric Group Theory. University of Chicago Press, Chicago, IL (2000)
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)
Duong, T.X., Ouhabaz, ElM., Sikora, A.: Plancherel type estimates and sharp spectral multipliers. J. Funct. Anal. 196, no. 2, 443–485 (2002)
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)
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)
Fukushima, M., Ōshima, Yō., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (1994)
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Springer-Verlag, Berlin (1990)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001)
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)
Grigor'yan, A.: Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10, no. 2, 395–452 (1994)
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)
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)
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)
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)
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)
Grigor'yan, A., Saloff-Coste, L.: Stability results for Harnack inequalities. Ann. Inst. Fourier (Grenoble) 55, no. 3, 825–890 (2005)
Grigor'yan, A., Saloff-Coste, L.: Heat kernel on manifolds with ends. (2007)
Grigor'yan, A., Telcs. A.: Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109, no. 3, 451–510 (2001)
Grigor'yan, A., Telcs. A.: Harnack inequalities and sub-Gaussian estimates for ran dom walks. Math. Ann. 324, no. 3, 521–556 (2002)
Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)
Guivarc'h, Y.: Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101, 333–379 (1973)
Gutiérrez, C.E., Wheeden, R.: Mean value and Harnack inequalities for degenerate parabolic equations. Colloq. Math. 60/61, no. 1, 157–194 (1990)
Gutiérrez, C.E., Wheeden, R.: Harnack's inequality for degenerate parabolic equa tions. Commun. Partial Differ. Equ. 16,no. 4–5, 745–770 (1991)
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)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. no. 688 (2000)
Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lect. Notes 5. Am. Math. Soc, Providence, RI (1999)
Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51,no. 5, 1437–1481 (2001)
Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York (2001)
Heinonen, J.: Nonsmooth calculus. Bull. Am. Math. Soc. (N.S.) 44, no. 2, 163–232 (electronic), (2007)
Kanai, M.: Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan 37, no. 3, 391–413 (1985)
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)
Kanai, M.: Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Japan 38, no. 2, 227–238 (1986)
Kleiner, B.: A new proof of Gromov's theorem on groups of polynomial growth. (2007) arXiv:0710.4593v4
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)
Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type Am. Math. Soc, Providence, RI (1968)
Levin, D., Solomyak, M.: The Rozenblum-Lieb-Cwikel inequality for Markov gener ators. J. Anal. Math. 71, 173–193 (1997)
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)
Li, P.: Harmonic sections of polynomial growth. Math. Res. Lett. 4, no. 1, 35–44 (1997)
Li, P., Yau, Sh.T.: On the Schrödinger equation and the eigenvalue problem. Com mun. Math. Phys. 88, no. 3, 309–318 (1983)
Li, P., Yau, Sh.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156,no. 3–4, 153–201 (1986)
Maz'ya, V.G.: Sobolev Spaces. Springer-Verlag, Berlin-Tokyo (1985)
Morrey, Ch.B., Jr.: Multiple Integrals in the Calculus of Variations. Springer-Verlag, New York (1966)
Moser, J.: On Harnack's theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)
Moser, J.: Correction to: “A Harnack inequality for parabolic differential equations.” Commun. Pure Appl. Math. 20, 231–236 (1967)
Moser, J.: On a pointwise estimate for parabolic differential equations. Commun. Pure Appl. Math. 24, 727–740 (1971)
Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Pivarski, M., Saloff-Coste, L.: Small time heat kernel behavior on riemannian com plexe
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)
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)
Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. 2, 27–38 (1992)
Saloff-Coste, L.: Parabolic Harnack inequality for divergence form second order dif ferential operators. Potential Anal.4, no. 4, 429–467 (1995)
Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. Cambridge Univ. Press, Cam bridge (2002)
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)
Semmes, S.: Some Novel Types of Fractal Geometry. Clarendon Press, Oxford (2001)
Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math.111, 247–302 (1964)
Sikora, A.: Sharp pointwise estimates on heat kernels. Q. J. Math., Oxf. II. Ser.47, no. 187, 371–382 (1996)
Sikora, A.: Riesz transform, Gaussian bounds and the method of wave equation. Math. Z.247, no. 3, 643–662 (2004)
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)
Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9)75, no. 3, 273–297 (1996)
Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness andL p-Liouville properties. J. Reine Angew. Math.456, 173–196 (1994)
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)
Sturm, K.T.: The geometric aspect of Dirichlet forms. In: New Directions in Dirichlet Forms, pp. 233–277. Am. Math. Soc., Providence, RI (1998)
Telcs, A.: The Art of Random Walks. Springer-Verlag, Berlin (2006)
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)
Varopoulos, N.Th.: Hardy-Littlewood theory for semigroups. J. Funct. Anal.63, no. 2, 240–260 (1985)
Varopoulos, N.Th.: Isoperimetric inequalities and Markov chains. J. Funct. Anal.63, no. 2, 215–239 (1985)
Varopoulos, N.Th.: Convolution powers on locally compact groups. Bull. Sci. Math. (2)111, no. 4, 333–342 (1987)
Varopoulos, N.Th., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Univ. Press, Cambridge (1992)
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)
Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Univ. Press, Cambridge (2000)
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Saloff-Coste, L. (2009). Sobolev Inequalities in Familiar and Unfamiliar Settings. In: Maz’ya, V. (eds) Sobolev Spaces In Mathematics I. International Mathematical Series, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85648-3_11
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