Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of sym bols globally defined on Rn+1 x Rn+1, and then develop a periodisation proc edure for the calculus of symbols on the cylinder T-n x R. We show Garding' s inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are neede d in localization arguments for the analysis of numerical approximation met hods.