m-Microlocal elliptic pseudodifferential operators acting on $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega })$

In the first part of the paper the authors study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fr'echet space $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega })$. In the second one non homogeneous microlocal properties are introduced and propagation of Sobolev singularities for solutions to (pseudo)differential equations is given. For both the arguments actual examples are provided.