Almost periodic pseudodifferential operators and Gevrey classes

Abstract

We study almost periodic pseudodifferential operators acting on almost periodic functions G s ap (R d ) of Gevrey regularity index s ≥ 1. We prove that almost periodic operators with symbols of Hörmander type S m ρ,δ satisfying an s-Gevrey condition are continuous on G s ap (R d ) provided 0 < ρ ≤ 1, δ = 0 and sρ ≥ 1. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic operators.,

Almost Periodic Functions and Their Applications: A Survey of Results and Perspectives

“…Almost periodic pseudo-differential operators and Gevrey classes: almost periodic pseudo-differential operators have been analyzed by numerous mathematicians including Coburn, Moyer, and Singer [206], Dedik [207], Iannacci, Bersani, Dell'Acqua, and Santucci [208], Pankov [209], Shubin [210][211][212][213], and Wahlberg [214]. In this part, we will present the main ideas and results of research study [215] by Oliaro, Rodino, and Wahlberg, only. It is well known that Shubin proved that almost periodic pseudodifferential operators act continuously on the space of smooth almost periodic functions as well as that the operator norm on L 2 equals that on the Hilbert space B 2 (R n ) of Besicovitch almost periodic functions whose Fourier coefficients are square summable.…”

“…In the papers of Shubin, some regularity results for formally hypoelliptic almost periodic pseudo-differential operators have been examined on the space W 2 − ∞ (R n ) ≔ ∪ t∈R W 2 t (R n ). In [215], the authors sought for ultradistributional analogues of the aforementioned results, working with almost periodic functions that are Gevrey regular of order s ≥ 1 (the difference between the real analytic case s � 1 and the pure ultradistributional case s > 1 should be emphasized here). If ∅ ≠ Ω ⊆ R n , then the space of all Gevrey functions of order s ≥ 1, denoted by G s (Ω), is defined as a collection of all infinitely differentiable functions F: R n ⟶ C such that, for each compact set K ⊆ R n , there exists a finite real constant C K > 0 such that |D α F(t)| ≤ C 1+|α| K α!…”

“…s for all t ∈ R n and α ∈ N n 0 . An instructive counterexample in the one-dimensional setting, with s > 1, is given in [215], Example 2.1, showing that this is not true in general: set g s (x) ≔ exp(−…”

“…We also have that F s ∈ G s (R) as well as that F s ∉ G s ap (R); see the notion explained in the following. Albeit not explicitly constructed in [215], it is our strong belief that this example can be transferred to the multidimensional setting without any serious difficulties, as well (more to the point, the case s � 1 has not been considered in [215], Example 2.1, and deserves further analyses).…”

Gabor systems and almost periodic functions

“…In connection with dynamical systems and differential equations they were studied by B. M. Levitan and V. V. Zhikov [15] starting from results of J. Favard. In the more general context of pseudo-differential calculus M. A. Shubin [21] introduced scales of Sobolev spaces of almost periodic functions and recently A. Oliaro, L. Rodino and P.Wahlberg [16] extended some of these results to almost periodic Gevrey classes.…”