Microlocal energy methods and pseudo-differential operators (Q801585)

It is well known that analyticity and positivity for Hermite kernels are deeply connected. For example, the Hermite positivity is preserved under analytic continuations. In this paper, we combine these facts with the theory of microfunctions and pseudodifferential operators. We give methods of estimation of microanalyticity via Hermite positive microkernels. Moreover, in those methods, the positivity can be weakened to the so-called quasipositivity; this is formulated by employing a class of Hermite positive pseudodifferential operators with infinite order. These methods are very much like energy methods, and so we believe that they have many applications to problems of pseudo-differential equations.