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

TL;DR

This paper investigates the boundary values of zero solutions to hypoelliptic differential operators within ultradistribution spaces, extending classical results and providing new proofs for existing theorems.

Contribution

It unifies and extends classical results on boundary values of solutions in ultradistribution spaces and offers new proofs for prior theorems by Langenbruch.

Findings

Unified boundary value theory for hypoelliptic operators

Extended classical results to ultradistribution spaces

Provided new proofs for existing boundary value theorems

Abstract

We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator $P(D) = P(D_x, D_t)$ on $\mathbb{R}^{d+1}$. 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 [23] about distributional boundary values of zero solutions of $P(D)$.