Boundedness for pseudodifferential operators on multivariate α-modulation spaces

Abstract

The α-modulation spaces M^s,α_{p, q}(R^d), 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 S^b_ρ,0 extends to a bounded operator σ(x, D): M^s,α_{p, q}(R^d)→M^s-b,α_{p, q}(R^d) provided 0≤α≤ρ≤1, and 1< p, q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class S^b_1,0 maps the Besov space B^s_{p, q}(R^d) into B^s-b_{p, q}(R^d).