On the anti-Wick symbol as a Gelfand-Shilov generalized function

Abstract

The purpose of this article is to prove that the anti-Wick symbol of an operator mapping S ( R n ) \mathcal {S}(\mathbb {R}^n) into S ′ ( R n ) \mathcal {S}’(\mathbb {R}^n) , which is generally not a tempered distribution, can still be defined as a Gel′fand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gel′fand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.