On bounded pseudodifferential operators in Wiener spaces

Abstract

We aim at extending the definition of the Weyl calculus to an infinite dimensional setting, by replacing the phase space R2n by B2, where (i,H,B) is an abstract Wiener space. A first approach is to generalize the integral definition using the Wigner function. The symbol is then a function defined on B2 and belonging to a L1 space for a Gaussian measure, the Weyl operator is defined as a quadratic form on a dense subspace of L2(B). For example, the symbol can be the stochastic extension on B2, in the sense of L. Gross, of a function F which is continuous and bounded on H2. In the second approach, this function F defined on H2 satisfies differentiability conditions analogous to the finite dimensional ones. One needs to introduce hybrid operators acting as Weyl operators on the variables of finite dimensional subset of H and as anti-Wick operators on the rest of the variables. The final Weyl operator is then defined as a limit and it is continuous on a L2 space. Under rather weak conditions, it is an extension of the operator defined by the first approach. We give examples of monomial symbols linking this construction to the classical pseudodifferential operators theory and other examples related to other fields or previous works on this subject.