Solutions fondamentales exactes
Beals, Richard
Journées équations aux dérivées partielles, (1998), p. 1-9 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 1998-01-01
@article{JEDP_1998____A1_0,
     author = {Beals, Richard},
     title = {Solutions fondamentales exactes},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {1998},
     pages = {1-9},
     zbl = {01808711},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/JEDP_1998____A1_0}
}

Beals, Richard. Solutions fondamentales exactes. Journées équations aux dérivées partielles,  (1998), pp. 1-9. http://gdmltest.u-ga.fr/item/JEDP_1998____A1_0/

[1] J. Aarão, A transport equation of mixed type, Dissertation, Yale University 1997. | Zbl 0920.35155

[2] M. S. Baouendi AND C. Goulaouic, Non-analytic hypoellipticity for some degenerate elliptic operators, Bull. Amer. Math. Soc. 78 (1972), 483-486. | MR 45 #5567 | Zbl 0276.35023

[3] R. Beals, A note on fundamental solutions, submitted. | Zbl 0923.35137

[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 97h:35029 | Zbl 0864.35005

[5] R. Beals, B. Gaveau, AND P. C. Greiner, Uniform hypoelliptic Green's functions, J. Math. Pures Appl., to appear. | Zbl 0915.35029

[6] R. Beals, B. Gaveau, P. C. Greiner, AND Y. Kannai, Exact fundamental solutions for a class of degenerate elliptic operators, in preparation. | Zbl 0933.35041

[7] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), 1-89. | MR 4,227e | Zbl 0061.46403

[8] G. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373-376. | MR 47 #3816 | Zbl 0256.35020

[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 51 #3719 | Zbl 0293.35012

[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 95m:32035 | Zbl 0848.32018

[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 57 #1574 | Zbl 0366.22010

[12] P. C. Greiner, A fundamental solution for a nonelliptic partial differential operator, Can. J. Math. 31 (1979), 1107-1120. | MR 81g:35096 | Zbl 0475.35003

[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 84d:35111 | Zbl 0434.35007

[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 54 #6298 | Zbl 0336.22007

[15] A. Klingler, New derivation of the Heisenberg kernel, Comm. PDE 22 (1997), 2051-2060. | MR 99h:35026 | Zbl 0896.58068

[16] A. N. Kolmogorov, Zufällige Bewegungen, Acta Math. 35 (1934), 116-117. | JFM 60.1159.01 | Zbl 0008.39906

[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 88f:42045 | Zbl 0607.35011