Boundary values of zero solutions of hypoelliptic differential operators in ultradistribution spaces

We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator P ( D ) = P ( D x , D t ) on R d + 1 . Our work unifies and considerably extends various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. We also give new proofs of several results of Langenbruch (Manuscripta Math. 26:17–35, 1978/79) about distributional boundary values of zero solutions of P ( D ).