Micro-local energy method of Mizohata and hypoellipticity (Q1112240)

The author studies hypoellipticity of the equation \(\partial u/\partial t+a(x,D_ x)u=f,\) where a(x,D) is a second order partial differential operator satisfying some estimates similar to Garding's inequality. He proves that for a solution \(u\in C'([0,\delta],{\mathcal D}'_ x)\), if \((x^ 0,\xi^ 0)\not\in WF(f(\cdot,t))\) for \(0\leq t\leq \delta\), then \((x^ 0,\xi^ 0)\not\in WF(u(\cdot,t))\) for \(0\leq t\leq \delta\). The main method in his proof is a microlocal energy method initiated by S. Mizohata in characterizing the analytic and Gevrey wave front sets.