Given a hypoelliptic boundary value problem on $\omega \times [0,T)$ with $\omega$ an open set in ${\mathbf{R}}^n$, $(n\>1)$, we show by matrix triangulation how to reduce it to two uncoupled first order systems, and how to estimate the eigenvalues of the corresponding matrices. Parametrices for the first order systems are constructed. We then characterize hypoellipticity up to the boundary in terms of the Calderon operator corresponding to the boundary value problem.

This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted $\partial W{F}_{\omega }\left(u\right)$ for distributions $u\in D^{\text{'}}({\mathbf{R}}_{+}^n)$, regular in the normal variable $x_n$ (thus, $\partial W{F}_{\omega }\left(u\right)=\varnothing$ means that $u\in {\cap }_{s+t=1/2}{H}^{s+t}$ near the boundary), and it is shown that $\partial W{F}_{\omega -m}\left[P\right(u^0{)}_{x_n\>0}]$ is a subset of $\partial WF\left(u\right)$ if $P$ has degree $m$ and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions of differential...