On the hypoellipticity of degenerate elliptic boundary value problems (Q1206975)

The main result of the paper is the Theorem: Let \(U\subset\mathbb{R}^ n\) be a neighborhood of the origin and let \(T>0\). Let \[ P=(\partial_ t-\mu(t,x)D_ x+b(t,x))(\partial_ t +\widetilde\mu(t,x)D_ x+ \widetilde b(t,x)), \] where \(\mu\), \(\widetilde \mu\), \(b\), \(\widetilde b\) are smooth functions on \([0,T]\times \overline{U}\). Assume \(\mu\widetilde\mu>0\) when \(t>0\) and either \(\mu(t,x)=t^ k a(t,x)\), \(\widetilde\mu(t,x)=t^ l \widetilde a(t,x)\) with \(k,l\geq 4\) or \(\mu(t,x)=\psi(t)a(t,x)\), \(\widetilde\mu(t,x)=\widetilde\psi(t)\widetilde a(t,x)\), where \(a\widetilde a>0\), when \(t=0\) and \(\psi\), \(\widetilde\psi\) are arbitrary smooth functions. If \(u\in C^ \infty([0,T],{\mathcal D}'(U))\) satisfies \[ Pu=f\in C^ \infty([0,T]\times U), \qquad u\mid_{t=0}=u_ 0\in C^ \infty(U), \] then \(u\in C^ \infty([0,T]\times U)\) perhaps after shrinking \(T\), \(U\). The theory of pseudo-differential operators is used for proving this theorem.