Bilinear Pseudo-differential Operators with Gevrey–Hörmander Symbols

We consider bilinear pseudo-differential operators whose symbols may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore, we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand–Shilov type spaces.