Boundedness for pseudodifferential operators on multivariate α-modulation spaces

Abstract

The α-modulation spaces Ms,αp, q(Rd), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x, D) with symbol in the Hörmander class Sbρ,0 extends to a bounded operator σ(x, D): Ms,αp, q(Rd)→Ms-b,αp, q(Rd) provided 0≤α≤ρ≤1, and 1< p, q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class Sb1,0 maps the Besov space Bsp, q(Rd) into Bs-bp, q(Rd).