Essential spectra of pseudodifferential operators in Sobolev spaces with variable smoothness and variable Lebesgue indices

A theorem about the boundedness of pseudodifferential operators (PDO) in Sobolev spaces with variable smoothness and variable Lebesgue indices is presented. The boundedness of PDOs in the Lebesgue spaces implies the boundedness of PDOs of Hörmander class. It is considered that PDOs whose symbols can be extended in the momentum variable over some tubular domain. The essential spectrum of a PDO does not depend on the functions, but it essentially depends on the exponential weight.