Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators
Autour de l'analyse microlocale - Volume en l'honneur de Jean-Michel Bony, Astérisque, no. 284 (2003), pp. 245-264.
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     author = {Morimoto, Yoshinori and Xu, Chao-Jiang},
     title = {Logarithmic {Sobolev} inequality and semi-linear {Dirichlet} problems for infinitely degenerate elliptic operators},
     booktitle = {Autour de l'analyse microlocale - Volume en l'honneur de Jean-Michel Bony},
     editor = {Lebeau Gilles},
     series = {Ast\'erisque},
     pages = {245--264},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {284},
     year = {2003},
     mrnumber = {2003422},
     zbl = {1096.35048},
     language = {en},
     url = {http://www.numdam.org/item/AST_2003__284__245_0/}
}
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Morimoto, Yoshinori; Xu, Chao-Jiang. Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators, in Autour de l'analyse microlocale - Volume en l'honneur de Jean-Michel Bony, Astérisque, no. 284 (2003), pp. 245-264. http://www.numdam.org/item/AST_2003__284__245_0/

[1] H. Ando and Y. Morimoto, Wick calculus and the Cauchy problem for some dispersive equations, to appear in Osaka J. Math., 39-1 (2002). | MR | Zbl

[2] J.-M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérées, Ann. Inst. Fourier, 19 (1969), 227-304. | EuDML | Numdam | MR | Zbl

[3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. | DOI | MR | Zbl

[4] J.-Y. Chemin and C.-J. Xu, Inclusions de Sobolev en calcul de Weyl-Hörmander et systèmes sous-elliptiques, Annales de l'École Normale Supérieure, 30 (1997), 719-751. | DOI | EuDML | Numdam | MR | Zbl

[5] M. Christ, Hypoellipticity in the infinitely degenerate regime, to appear in proceedings of Ohio State university conference on several complex variable. | MR | Zbl

[6] M. Derridj, Un problème aux limites pour une classe d'opérateurs du second ordre hypoelliptiques, Annales de l'Institut Fourier 21 (1971), 99-148. | DOI | EuDML | Numdam | MR | Zbl

[7] B. Franchi, S. Gallot and R. I. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Mathematische Annalen, 300 (1994), 557-571. | DOI | EuDML | MR | Zbl

[8] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd ed.; Springer-Verlag, Berlin-Now York, 1983. | MR | Zbl

[9] D. Jerison, The Dirichlet problem for the Kohn-Laplacian on the Heisenberg group, Parts I and II, J. Funct. Analysis, 43 (1981), 97-142. | DOI | MR | Zbl

[10] M. Koike, A note on hypoellipticity for degenerate elliptic operators, Publ. RIMS Kyoto Univ., 27 (1991), 995-1000. | DOI | MR | Zbl

[11] J. J. Kohn, Pseudo-differential operators and non-elliptic problem, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968), Edizioni Cremonese, Rome (1969), 157-165. | MR

[12] J. J. Kohn, Hypoellipticity at points of infinite type, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, 1998), Contemp. Math., 251 (2000), 393-398. | MR | Zbl

[13] N. Lerner, The Wick calculus of pseudo-differential operators and energy estimates, "New trends in microlocal analysis" (J.-M. Bony and M. Morimoto, eds.), Springer- Verlag, Berlin, Heiderberg, New York, Tokyo (1996), 23-37. | MR | Zbl

[14] Y. Morimoto and T. Morioka, The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators, Bull. Sc. Math. 121 (1997), 507-547. | MR | Zbl

[15] Y. Morimoto and T. Morioka, Hypoellipticity for elliptic operators with infinite degeneracy, "Partial Differential Equations and Their Applications" (Chen Hua and L. Rodino, eds.), World Sci. Publishing, River Edge, NJ, (1999), 240-259. | MR | Zbl

[16] O. A. Olejnik and E. V. Radkevic, Second order equations with non-negative characterisitic form, Plenum Press, New York London, 1973. | MR

[17] O. S. Rothaus, Lower bounds for eigenvalues of regular Strum-Liouville operators and the logarithmic Sobolev inequality, Duke Math. J., 45 (1978), 351-362. | DOI | MR | Zbl

[18] Sawyer E., A weighted inequality and eigenvalue estimates for Schrödinger operators, Indiana Univ. Math. J., 35 (1986), 1-28. | DOI | MR | Zbl

[19] E. M. Stein, Note on the class L log L, Studia, Math. 32 (1969), 305-310. | DOI | EuDML | MR | Zbl

[20] Trudinger N. S., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Sup. Pisa, 22 (1968), 265-274. | EuDML | Numdam | MR | Zbl

[21] Wakabayashi S. and Suzuki M., Microhypoellipticity for a class of pseudo-differential operators with double characteristics, Funkciaj Ekvacioj, 36 (1993), 519-556. | MR | Zbl

[22] C.-J. Xu, Subelliptic variational problems, Bull. Soc. Math. France 118 (1990), 147-169. | DOI | EuDML | Numdam | MR | Zbl

[23] C.-J. Xu, Regularity problem for quasi-linear second order subelliptic equations, Comm. Pure Appl. Math., 45 (1992), 77-96. | DOI | MR | Zbl

[24] C.-J. Xu, Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmander's condition, Chinese J. Contemp. Math., 15 (1994), 183-193. | MR | Zbl