Solutions fondamentales exactes
Journées équations aux dérivées partielles (1998), article no. 1, 9 p.

Exact fundamental solutions are known for operators of various types. We indicate a general approach that gives various old and new fundamental solutions for operators with double characteristics. The solutions allow one to read off detailed behavior, such as the presence or absence of analytic hypoellipticity. Recent results for operators with multiple characteristics are also described.

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     author = {Beals, Richard},
     title = {Solutions fondamentales exactes},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {1},
     pages = {1--9},
     publisher = {Universit\'e de Nantes},
     year = {1998},
     zbl = {01808711},
     language = {fr},
     url = {http://www.numdam.org/item/JEDP_1998____A1_0/}
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Beals, Richard. Solutions fondamentales exactes. Journées équations aux dérivées partielles (1998), article  no. 1, 9 p. http://www.numdam.org/item/JEDP_1998____A1_0/

[1] J. Aarão, A transport equation of mixed type, Dissertation, Yale University 1997. | Zbl

[2] M. S. Baouendi AND C. Goulaouic, Non-analytic hypoellipticity for some degenerate elliptic operators, Bull. Amer. Math. Soc. 78 (1972), 483-486. | MR | Zbl

[3] R. Beals, A note on fundamental solutions, submitted. | Zbl

[4] R. Beals, B. Gaveau, AND P. C. Greiner, On a geometric formula for the fundamental solutions of subelliptic laplacians, Math. Nachrichten 181 (1996), 81-163. | MR | Zbl

[5] R. Beals, B. Gaveau, AND P. C. Greiner, Uniform hypoelliptic Green's functions, J. Math. Pures Appl., to appear. | Zbl

[6] R. Beals, B. Gaveau, P. C. Greiner, AND Y. Kannai, Exact fundamental solutions for a class of degenerate elliptic operators, in preparation. | Zbl

[7] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), 1-89. | MR | Zbl

[8] G. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373-376. | MR | Zbl

[9] G. Folland AND E. M. Stein, Estimates for the ∂b-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. | MR | Zbl

[10] G. Francsics AND N. Hanges, Explicit formulas for the Szegö kernel on certain weakly pseudoconvex domains, Proc. Amer. Math. Soc. 123 (1995), 3161-3168. | MR | Zbl

[11] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95-153. | MR | Zbl

[12] P. C. Greiner, A fundamental solution for a nonelliptic partial differential operator, Can. J. Math. 31 (1979), 1107-1120. | MR | Zbl

[13] P. C. Greiner AND E. M. Stein, On the solvability of some differential operators of type ʬb, Ann. Scuola Norm. Pisa Cl. Sci. 4 (1978), 106-165. | MR | Zbl

[14] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math. 56 (1976), 165-173. | MR | Zbl

[15] A. Klingler, New derivation of the Heisenberg kernel, Comm. PDE 22 (1997), 2051-2060. | MR | Zbl

[16] A. N. Kolmogorov, Zufällige Bewegungen, Acta Math. 35 (1934), 116-117. | JFM | Zbl

[17] A. Nagel, Vector fields and nonisotropic metrics. in «Beijing Lectures in Harmonic Analysis,» Annals of Math. Studies 112, Princeton Univ. Press, Princeton 1986, pp. 241-306. | MR | Zbl