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 oper-, 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 ).

Continuous Frames, Function Spaces, and the Discretization Problem

“…The α-modulation spaces are known to have nice analysis properties. For instance, the mapping properties of pseudodifferential operators in Hörmander classes on α-modulation spaces as by Holschneider, Nazaret [42] and Borup [7] generalize classical results of Cordoba and Fefferman [10]. Moreover, we expect that such spaces have a key role in the study of pseudodifferential operators modeling the transmission of (digital) signals in wireless communications and in corresponding numerical methods [11].…”

Frame Decomposition of Decomposition Spaces

“…Another important application of discrete decompositions of spaces is to simplify the analysis of operators acting on it. Pseudo-differential and Fourier integral operators on Besov and modulation spaces have been studied extensively, see [2,6,8,10,11,32,39,41,44,45] and references herein.…”

Nonlinear approximation in α ‐modulation spaces

“…In [3] pseudodifferential operators on α-modulation spaces are studied. Denote by OPS s ρ,δ , s ∈ R, ρ ∈ (0, 1], and δ ∈ [0, 1), the class of pseudodifferential operators with symbols in the Hörmander class S s ρ,δ , see [3] for the definition.…”

“…In [3] the results in the present paper are used to study pseudodifferential operators on α-modulation spaces. It is proven that certain pseudodifferential operators with symbols of Hörmander type extends naturally to bounded operators on α-modulation spaces.…”