Laurent series expansion for solutions of hypoelliptic equations

M. Langenbruch¹

¹ Department of Mathematics University of Oldenburg D-26111 Oldenburg, Germany

Annales Polonici Mathematici, Tome 78 (2002) no. 3, pp. 277-289.

Résumé

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 $1$.

Détail

DOI : 10.4064/ap78-3-6

Mots-clés : prove zero solution hypoelliptic partial differential operator expanded generalized laurent series near point singularity only operator semielliptic moreover coefficients may calculated means cauchy integral type formula particular obtain explicit expansions solutions heat equation near point singularity prove necessity semiellipticity additionally assume index hypoellipticity respect variable