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

Commun. Math. Anal.
Volume 10, Number 2 (2011), 75 - 96

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

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Abstract

We consider the invertibility of parabolic pseudodifferential operators in exponential weighted Sobolev spaces. We suppose that the symbol aof the operator Op(a) is analytically extended with respect to the impulse variable in an unbounded tube domain Rn+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.