Let Phi be a Young function. We study convolution properties for symbol classes s_A Phi, which consist of all a such that the pseudo-differential operator OpA(a) is in the Orlicz Schatten class I Phi(L²(R^d)). Especially we prove Young type results for such classes. We apply the results to Toeplitz operators and prove Orlicz Schatten properties of such operators.

We consider a broad class of modulation spaces M(ω, B),parameterized with weight function ω and a normal quasiBanach function space B of order r0 ∈ (0, 1]. Then we provethat f ∈ M(ω, B), if and only if Vϕf belongs to the Wieneramalgam space Wr(ω, B), and

∥f∥M(ω,B) ≍ ∥Vϕf · ω∥B ≍ ∥Vϕf∥Wr(ω,B), r ∈ [r0, ∞].

We use the results to extend and improve continuity andlifting properties for pseudo-differential and Toeplitz operatorswith symbols in weighted M∞,r0 -spaces, r0 ≤ 1, when actingon M(ω, B)-spaces.

We study structural properties of WLg,theta q,p, which are Wiener-Lebesgue spaces with respect to a slowly varying metric g and with parameters p, q is an element of (0,infinity], theta is an element of R. For p is an element of (0, 1], we deduce Schatten-p properties for pseudo-differential operators whose symbols, together with their derivatives, obey suitable WLg,theta q,p-boundedness conditions. Especially, we perform such investigations for the Weyl-Hormander calculus. Finally, we apply our results to global-type SG and Shubin pseudo-differential operators.

Let G be a compact Lie group of dimension n. In this work we characterise the membership of classical pseudo-differential operators on G in the trace class ideal & Sscr;(1)(L-2(G)) , as well as in the setting of the Schatten ideals & Sscr;(r) (L-2(G)) , for all r > 0. In particular, we deduce Schatten characterizations of elliptic pseudo-differential operators of (rho,delta)-type for the large range 0 <= delta < rho <= 1. Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space G x G, with the momentum variables belonging to the unitary dual G of G. In terms of the parameters ( rho , delta ), on the torus T (n), we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class Psi(-& varkappa;)(0,0)(T-n), kappa > 0, belonging to all the Schatten ideals. Additional order criteria are given in the setting of classical pseudo-differential operators. We present also some open problems in this setting.

We deduce factorization properties for Wiener amalgam spaces WLp,q, an extended family of modulation spaces M(omega, B), and for Schatten symbols s(p)(w) in pseudo-differential calculus under e. g. convolutions, twisted convolutions and symbolic products. Here M(omega, B) can be any quasi-Banach Orlicz modulation space. For example we show that WL1,r * WLp,q = WLp,q and WL1,r#s(p)(w) = s(p)(w) when r is an element of (0, 1], r _= p, q _ infinity. In particular we improve Rudin's identity L-1 * L-1 = L-1. (c) 2024 The Author(s). Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG).

We establish Hermite expansion characterizations for several subspaces of the Fréchet space of functions on the real line satisfying |f(x)|≲e−(1/2−λ)x², |fˆ(ξ)|≲e−(1/2−λ)ξ², ∀λ>0. In particular, we extend and improve Fourier characterizations of the so-called proper Pilipović spaces obtained in [21]. The main ingredients in our proofs are the Bargmann transform and some achieved optimal forms of the Phragmén-Lindelöf principle.

Let H be a Hilbert space, continuously embedded in S'(R^d), and which contains at least one non-zero element in S'(R^d). If there is a constant C₀ > 0 such that ||e^{i❮· ,ε❯} f ( ·  -x)||_H ≤ C₀||f||_H, f ∈ H, x, ε ∈ R^d, then we prove that H = L²(R^d), with equivalent norms.

We deduce continuity and (global) wave-front properties of classes of Fourier multipliers, pseudo-differential, and Fourier integral operators when acting on Orlicz spaces, or more generally, on Orlicz–Sobolev type spaces. In particular, we extend Hörmander’s improvement of Mihlin’s Fourier multiplier theorem to the framework of Orlicz spaces. We also show how Young functions Φ of the Orlicz spaces are linked to properties of certain Lebesgue exponents pΦ and qΦ emerged from Φ.

We deduce continuity properties for pseudo-differential operators with symbols in Orlicz modulation spaces when acting on other Orlicz modulation spaces. In particular we extend well-known results in the literature. For example we generalize the classical result by Cordero and Nicola that if [Formula presented], pj,qj⩽q′,q⩽p and a∈Mp,q, then the pseudo-differential operator Op(a) is continuous from Mp to Mp. We also show that the entropy functional Eϕ possess suitable continuity properties on a suitable Orlicz modulation space MΦ satisfying Mp⊆MΦ⊆M2, though Eϕ is discontinuous on M2=L2. 

We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1. As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sj & ouml;strand's results.