\(\mathrm{m}\)-microlocal elliptic pseudodifferential operators acting on \(L_{\mathrm{loc}}^p(\Omega)\) (Q2830667)

In the present article the authors consider a modified theory of the standard pseudodifferential operators. They use a class of local vector-weighted symbols instead of Hörmander local symbol classes. The corresponding pseudodifferential operators are called \(m\)-pseudodifferential operators, which could be considered in the frame of general pseudodifferential calculus of R. Beals and L. Hörmander.NEWLINENEWLINEIn the first part of paper the minimal and maximal extension of \(m\)-pseudodifferential operators in the Fréchet space \(L^p_{loc}(\Omega)\) is constructed. It is shown that the minimal and maximal extensions coincide for elliptic (in generalized sense) properly supported \(m\)-pseudodifferential operators and they have as domain weighted local Sobolev space. In the second part of the article, the concepts of characteristic filter \(m\)-pseudodifferential operator and of filter of weighted Sobolev regularity are introduced, and then a microlocal propagation of Sobolev singularities for solutions to (pseudo)differential equations is obtained.