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An extension of Hörmander’s theorem for infinitely degenerate second-order operators. (English) Zbl 0840.60053

The hypoellipticity of a class of second-order operators \(L\) which do not satisfy Hörmander’s conditions is proved. More precisely, it is assumed that the set of points where these conditions do not hold is contained in a submanifold of codimension 1; then the result is proved under another assumption concerning the behaviour near this submanifold. The parabolic equation associated to the operator \(L + \partial/ \partial t\) is also studied. The proofs of these results are probabilistic, and are based on Itô’s and Malliavin’s calculus.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
35J15 Second-order elliptic equations
35K10 Second-order parabolic equations
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[1] D. R. Bell, The Malliavin calculus , Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 34, Longman Scientific & Technical, Harlow, 1987. · Zbl 0678.60042
[2] D. R. Bell and S.-E. A. Mohammed, Hypoelliptic parabolic operators with exponential degeneracies , C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 11, 1059-1064. · Zbl 0790.35016
[3] C. Fefferman and D. Phong, Subelliptic eigenvalue problems , Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590-606. · Zbl 0503.35071
[4] L. Hörmander, Hypoelliptic second order differential equations , Acta Math. 119 (1967), 147-171. · Zbl 0156.10701 · doi:10.1007/BF02392081
[5] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes , North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam, 1989. · Zbl 0684.60040
[6] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. II , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 1-76. · Zbl 0568.60059
[7] P. Malliavin, Stochastic calculus of variation and hypoelliptic operators , Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), Wiley, New York, 1978, pp. 195-263. · Zbl 0411.60060
[8] P. Malliavin, \(C\spk\)-hypoellipticity with degeneracy. II , Stochastic analysis (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1978), Academic Press, New York, 1978, pp. 327-340. · Zbl 0449.58023
[9] Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators , Osaka J. Math. 24 (1987), no. 1, 13-35. · Zbl 0658.35039
[10] J. Norris, Simplified Malliavin calculus , Séminaire de Probabilités, XX, 1984/85, Lecture Notes in Math., vol. 1204, Springer, Berlin, 1986, pp. 101-130. · Zbl 0609.60066 · doi:10.1007/BFb0075716
[11] O. A. Oleĭ nik and E. V. Radkevič, Second order equations with nonnegative characteristic form , Plenum Press, New York, 1973.
[12] D. Stroock, The Malliavin calculus, a functional analytic approach , J. Funct. Anal. 44 (1981), no. 2, 212-257. · Zbl 0475.60060 · doi:10.1016/0022-1236(81)90011-2
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