Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey type pseudo-differential operators

The Weyl product on quasi-Banach modulation spaces

“…where Op A (a) is the A-indexed pseudodifferential operator with symbol a. This family of calculi contains the Weyl quantization as the special case A = 1 2 I. The sufficient conditions and the necessary conditions that we find extend results [7,24] where the same problem was studied for the narrower range of Lebesgue parameters [1, ∞].…”

“…[29,Proposition 2.6]. ) On one hand we have f M p,q c ℓ q for any p > 0 due to Proposition 1.6 (1). Thus f ∈ p>0 M p,q (R d ).…”

Bilinear pseudo-differential operators with Gevrey-Hörmander symbols

“…x, ξ, η, x 1 , ξ 1 , η 1 ∈ R d , for some constants h, R > 0. This, together with the Leibnitz rule applied to ∂ α x ∂ β ξ ∂ γ η e i( x,ζ + y,ξ + z,η ) V φ a(x, ξ, η, ζ, y, z) gives (2).…”

“…Modulation spaces, originally introduced by Feichtinger in [16], are recognized as appropriate family of spaces when dealing with problems of time-frequency analysis, see [16-20, 24, 35,37], to mention just a few references. A broader family of modulation spaces is recently studied in [2,33,49]. Let s, σ > 0, such that s + σ ≥ 1, and let φ ∈ S σ s (R d ) be fixed.…”

“…In this paper, we employ the techniques of time-frequency analysis and modulation spaces, and consider bilinear pseudo-differential operators of Gevrey-Hörmander type whose symbols are of infinite order and may have a (super-)exponential growth at infinity, together with all their derivatives. The linear counterpart of such operators is considered in [9], within the environment of isotropic Gelfand-Shilov spaces of functions and distributions, see also [2,49], and extended to the anisotropic setting in [1,3]. The main purpose of this paper is to extend some boundedness results given there to the bilinear case.…”