Lp-Bounded Pseudodifferential Operators and Regularity for Multi-quasi-elliptic Equations

Abstract

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.

m‐Microlocal elliptic pseudodifferential operators acting on

“…XI, Prop. 4.5], see also [, Theorem 4.1], the next result immediately follows: Theorem If , then, for any : If is assumed to be properly supported, then it is bounded both as operator on and on .…”

“…More precisely , where are the vertices of a complete Newton polyhedron. Roughly speaking the normal vector to any face of not lying on the coordinate axes has not zero components, see [, §1.1], . Moreover again from [, §1.1], we have that the formal order of is .…”

“…x 2 is clearly M-elliptic, as operator in S · M , M (R 2 ), while the Schrödinger operator i∂ x 1 − ∂ 2 , where V (P) = (0, 0), (3, 0), (2, 2), (0, 3) are the vertices of a complete Newton polyhedron. Roughly speaking the normal vector to any face of P not lying on the coordinate axes has not zero components, see [5, §1.1], [10]. Moreover again from [5, §1.1], we have that the formal order of λ(ξ ) is μ = 6.…”

“…The corresponding pseudodifferential operators will be called in the following m ‐pseudodifferential operators and they could be considered in the frame of general pseudodifferential calculus of R. Beals , Hörmander . Thanks to the assumptions on the weight vector , the classes satisfy a condition of Taylor type and the m ‐pseudodifferential operators are continuous when the weight is bounded, see and the next Theorem .…”

$$L^p$$ Continuity and Microlocal Properties for Pseudodifferential Operators

“…(cf. [12], Proposition 3.1). The usual symbolic calculus of pseudodifferential operators holds true in Op M m ρ .…”