On the Invertibility of Parabolic Pseudodifferential Operators in General Exponential Weighted Spaces

Abstract

We consider the invertibility of parabolic pseudodifferential operators in exponential weighted Sobolev spaces. We suppose that the symbol $a$ of the operator $Op(a)$ is analytically extended with respect to the impulse variable in an unbounded tube domain $\mathbb{R}^{n}+iD$ and satisfies conditions of uniform parabolicity . We prove that under these conditions the pseudodifferential operator $Op(a)$ is invertible in admissible weighted Sobolev spaces with weights connected with the domain $D.$ As an application we obtain exponential estimates of solutions (including estimates of the fundamental solution) for parabolic differential operators.