Commun. Math. Anal.

Volume 10, Number 2 (2011), 75 - 96

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

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 *O**p*(*a*) is analytically extended with respect to the impulse variable in an unbounded tube domain R*n*+*i**D* and satisfies conditions of uniform parabolicity . We prove that under these conditions the pseudodifferential operator *O**p*(*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.