Laurent series expansion for solutions of hypoelliptic equations
Volume 78 / 2002
Annales Polonici Mathematici 78 (2002), 277-289 MSC: Primary 35H10, 35C10; Secondary 35E20 DOI: 10.4064/ap78-3-6
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 $1$.