We prove that any zero solution of a hypoelliptic partial differential operator can be expanded in a generalized Laurent series near a point singularity if and only if the operator is semielliptic. Moreover, the coefficients may be calculated by means of a Cauchy integral type formula. In particular, we obtain explicit expansions for the solutions of the heat equation near a point singularity. To prove the necessity of semiellipticity, we additionally assume that the index of hypoellipticity with respect to some variable is 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap78-3-6,
author = {M. Langenbruch},
title = {Laurent series expansion for solutions of hypoelliptic equations},
journal = {Annales Polonici Mathematici},
volume = {79},
year = {2002},
pages = {277-289},
zbl = {0988.35045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap78-3-6}
}
M. Langenbruch. Laurent series expansion for solutions of hypoelliptic equations. Annales Polonici Mathematici, Tome 79 (2002) pp. 277-289. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap78-3-6/