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).

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.

Let f be a function or distribution on Rd. We characterize Pilipovic space in terms of certain estimates of the involved functions and suitable choices of their fractional Fourier transforms. For the analysis we derive a multi-dimensional version of Phragmen-Lindelof's theorem.

We give a proof of that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties and fix time estimates for such operators on modulation spaces, and apply the results to prove Strichartz estimates for such propagators when acting on Pilipovic and modulation spaces. Especially we extend some results by Balhara, Cordero, Nicola, Rodino and Thangavelu. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.

We deduce continuity properties for pseudo-differential operators with symbols in quasi-Banach Orlicz modulation spaces when rely on other quasi-Banach Orlicz modulation spaces. In particular we extend some earlier results.

We deduce trace properties for modulation spaces (including certain Wiener-amalgam spaces) of Gelfand-Shilov distributions.We use these results to show that psi dos with amplitudes in suitable modulation spaces, agree with normal type psi dos whose symbols belong to (other) modulation spaces. In particular we extend earlier trace results for modulation spaces, to include quasi-Banach modulation spaces. We also apply our results to extend earlier results on Schatten-von Neumann and nuclear properties for psi dos with amplitudes in modulation spaces.

We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schrödinger equation.