Continuity properties in non-commutative convolution algebras, with applications in pseudo-differential calculus

Abstract

We study continuity properties for a family { s p } p⩾1 of increasing Banach algebras under the twisted convolution, which also satisfies that a∈ s p , if and only if the Weyl operator a w ( x, D) is a Schatten–von Neumann operator of order p on L 2. We discuss inclusion relations between the s p -spaces, Besov spaces and Sobolev spaces. We prove also a Young type result on s p for dilated convolution. As an application we prove that f( a)∈ s 1, when a∈ s 1 and f is an entire odd function. We finally apply the results on Toeplitz operators and prove that we may extend the definition for such operators.