Pseudolocality and microlocality of general classes of pseudodifferential operators

We study the pseudolocality and microlocality of pseudodifferential operato rs with general symbols. We treat the problem of pseudo- and microlocality, along finite dimensional linear subspaces, of the Weyl quantization and al so generalize some results of PARENTI and RODINO to the case of the Weyl-Ho rmander calculus of pseudodifferential operators. We do not require that th e metric, in the definition of the symbol, is decomposable, sigma temperate or satisfy the uncertainty principle neither the weight function of the sy mbol has to be sigma -g temperate. Instead, we assume weaker conditions on the metric and the weight function and the only extra condition we are assu ming on the symbol is one of its essential support.