Using a classical result of Marcinkiewicz and Lizorkin about the L p -continuity for Fourier multipliers, the authors study the action of a class of pseudodifferential operators with weighted smooth symbol on a family of weighted Sobolev spaces. Results about L p -regularity for multi-quasi-elliptic pseudodifferential operators are also given.
In the first part of the paper we study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space Lloc p(Ω). 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
We obtain a result of continuity for pseudodifferential operator whose symbols belong to fully inhomogeneous weighted Sobolev spaces, which realize moreover to be algebras with respect to the pointwise multiplication. Applications to the study of the local regularity for some classes of non linear partial differential equations are given, besides some remarks about a number of weight functions which appear in the literature related to the weighted pseudodifferential calculus.
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