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A Fundamental Solution for a Nonelliptic Partial Differential Operator

Published online by Cambridge University Press:  20 November 2018

Peter C. Greiner
Peter C. Greiner*
Affiliation:
University of Toronto, Toronto, Ontario
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Let

(1)

and set

(2)

Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”

(3)

In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 1107 - 1120
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Greiner, P. C. and Stein, E. M., Estimates for the d-Neumann problem, Mathematical Notes Series, 19 (Princeton Univ. Press, Princeton, N.J., 1977).Google Scholar
2. Greiner, P. C. and Stein, E. M., On the solvability of some differential operators of type \b, Seminar on Several Complex Variables, Cortona, Italy, 1976.Google Scholar
3. Hôrmander, L., Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.Google Scholar
4. Kohn, J. J., Harmonic integrals for differential complexes, Global Analysis, Princeton Math. Series, 29 (Princeton Univ. Press, Princeton, N.J., 1969), 295–308.Google Scholar
5. Rothschild, L. P. and Stein, E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247–320.Google Scholar