Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms

Kaplan, Aroldo

doi:10.1090/S0002-9947-1980-0554324-X

Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms
HTML articles powered by AMS MathViewer

by Aroldo Kaplan PDF

Trans. Amer. Math. Soc. 258 (1980), 147-153 Request permission

Abstract:

We introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group.

References

J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632. MR 139178, DOI 10.2307/1970213
M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl, suppl. 1, 3–38. MR 167985, DOI 10.1016/0040-9383(64)90003-5

Beweis des Satzes von Hurwitz-Radon

15

G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376. MR 315267, DOI 10.1090/S0002-9904-1973-13171-4
G. B. Folland, On the Rothschild-Stein lifting theorem, Comm. Partial Differential Equations 2 (1977), no. 2, 165–191. MR 433514, DOI 10.1080/03605307708820028
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
Aroldo Kaplan and Robert Putz, Boundary behavior of harmonic forms on a rank one symmetric space, Trans. Amer. Math. Soc. 231 (1977), no. 2, 369–384. MR 477174, DOI 10.1090/S0002-9947-1977-0477174-1
T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR 0396410
Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
François Trèves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline \partial$-Neumann problem, Comm. Partial Differential Equations 3 (1978), no. 6-7, 475–642. MR 492802, DOI 10.1080/03605307808820074