Pseudo-differential operators in a Gelfand-Shilov setting

Abstract

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand–Shilov spaces. Moreover, we deduce composition and certain invariance properties of these classes.

Pseudo-differential calculus in a Bargmann setting

“…Here we note that the operator e ixAD ξ ,Dxy is homeomorphic on Σ 1 pR 2d q and its dual (cf. [5,6,29]). For modulation spaces we have the following subresult of Proposition 2.8 in [28].…”

“…Then the condition on a in Theorem 3.8 is that }V φ 0 a¨ω 0 } L p,q ă 8. In view of [5,6,29] and Proposition 1.13 (2), the previous condition is the same as }V φ A a¨ω 0 } L p,q ă 8 because ω 0 is moderate. We observe that all weights in Theorem 3.8 are moderate, while there are no such assumptions or other restrictions on the involved weight functions in Theorem 3.3.…”

Factorizations and Singular Value Estimates of Operators with Gelfand–Shilov and Pilipović kernels

“…The product a# A b is well-defined and is uniquely extendable in different ways (see e. g. [1,4,16]). …”

“…4. More precisely, an operator class M is called a factorization algebra, if for every T ∈ M, there exist T 1 , T 2 ∈ M such that T = T 1 • T 2 .…”

“…4 we extend the latter property such that all Pilipović spaces are included. That is, we prove that the set of operators with kernels in a fixed (but arbitrarily chosen) Pilipović space is a factorization algebra.…”