Wiener Property Research Articles

Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik (Ann Inst Fourier 58:2279–2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219–233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrödinger equations, arXiv:2208.00505).

Paley–Wiener properties for spaces of power series expansions

ABSTRACT We extend Paley–Wiener results in the Bargmann setting deduced in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.] to larger classes of power series expansions. At the same time, we deduce characterizations of all Pilipović spaces and their distributions (and not only of low orders as in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.]).

Almost Diagonalization of $$\tau $$ τ -Pseudodifferential Operators with Symbols in Wiener Amalgam and Modulation Spaces

In this paper we focus on the almost-diagonalization properties of $\tau$-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sj\"ostrand class, for which Gr\"ochenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any $\tau$-pseudodifferential operator, for $\tau \in [0,1]$, also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for $\tau$-pseudodifferential operators on Wiener amalgam and modulation spaces.

Twisted Orlicz algebras, I

Let G be a locally compact group, let $\Omega:G\times G\to \mathbb{C}^*$ be a 2-cocycle, and let $\Phi$ be a Young function. In this paper, we consider the Orlicz space $L^\Phi(G)$ and investigate its algebraic property under the twisted convolution $\circledast$ coming from $\Omega$. We find sufficient conditions under which $(L^\Phi(G),\circledast)$ becomes a Banach algebra or a Banach $*$-algebra; we call it a {\it twisted Orlicz algebra}. Furthermore, we study its harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, and having Wiener property, mostly for the case when $G$ is a compactly generated group of polynomial growth. We apply our methods to several important classes of polynomial as well as subexponential weights and demonstrate that our results could be applied to variety of cases.

Absence of two Paley–Wiener properties for semisimple Lie groups of real rank one

The weak Paley–Wiener property and the topological Paley–Wiener property for connected semisimple Lie groups of real rank one with finite center are discussed.

Harmonic analysis of weighted [formula omitted]-algebras

Let G be a locally compact, compactly generated group of polynomial growth and let ω be a weight on G. Under proper assumptions on the weight ω, the Banach space Lp(G,ω) is a Banach ∗-algebra. In this paper we give examples of such weighted Lp-algebras and we study some of their harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, weak Wiener property, Wiener property, and existence of minimal ideals of a given hull.

IDEALS IN OPERATOR SPACE PROJECTIVE TENSOR PRODUCT OF C*-ALGEBRAS

AbstractLet A and B be C*-algebras. We prove the slice map conjecture for ideals in the operator space projective tensor product $A \mathbin {\widehat {\otimes }} B$. As an application, a characterization of the prime ideals in the Banach *-algebra $A\mathbin {\widehat {\otimes }} B$ is obtained. In addition, we study the primitive ideals, modular ideals and the maximal modular ideals of $A\mathbin {\widehat {\otimes }} B$. We also show that the Banach *-algebra $A\mathbin {\widehat {\otimes }} B$ possesses the Wiener property and that, for a subhomogeneous C*-algebra A, the Banach * -algebra $A \mathbin {\widehat {\otimes }} B$ is symmetric.

Failure of Wiener's property for positive definite periodic functions

We say that Wiener's property holds for the exponent p > 0 whenever a positive definite function f, which belongs to L p ( − ε , ε ) for some ε > 0 , necessarily belongs to L p ( T ) , too. This holds true for p ∈ 2 N by a classical result of Wiener. Recently various concentration results were proved for idempotents and positive definite functions on measurable sets on the torus. They enable us to prove a sharp version of the failure of Wiener's property for p ∉ 2 N , strengthening results of Wainger and Shapiro. To cite this article: A. Bonami, S.Gy. Révész, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Functional Calculus in Weighted Group Algebras

Let G be a compactly generated, locally compact group with polynomial growth and let ! be a weight on G. We look for general conditions on the weight which allow us to develop a functional calculus on a total part of L 1 (G,!). This functional calculus is then used to study harmonic analysis properties of L 1 (G,!), such as the Wiener property and Domar’s theorem.

Weighted group algebras on groups of polynomial growth

Let G be a compactly generated group of polynomial growth and ω a weight function on G. For a large class of weights we characterize symmetry of the weighted group algebra L 1 (G,ω). In particular, if the weight ω is sub-exponential, then the algebra L 1 (G,ω) is symmetric. For these weights we develop a functional calculus on a total part of L 1 (G,ω) and use it to prove the Wiener property.

Weak Paley–Wiener property for completely solvable Lie groups

Weak Paley–Wiener property for completely solvable Lie groups

Ideal structure of Beurling algebras on [ FC] − groups

It is proved that for an [ FC] − group G, the Beurling algebra L ω 1( G) is ∗-regular if and only if ω is non-quasianalytic. As an application the Wiener property is deduced. Further for a σ-compact [ FD] − group G, points in Prim ∗ L ω 1( G) are shown to be spectral provided that ω satisfies Shilov's conditions.

A characterization of groups with the one-sided Wiener property.

A characterization of groups with the one-sided Wiener property.

On Onesided Harmonic Analysis

It is shown that, in locally compact groups satisfying the onesided version of the Wiener property introduced by Leptin, characters of closed subgroups have always continuous positive definite extensions onto the whole group. We give a quick proof that for such groups the connected component of the identity is the direct product of a compact group and a vector group. In (4], Leptin considers the following onesided version of Wiener's theorem for a locally compact group G. (L) Every closed proper left ideal in L'(G) is annihilated by nonzero positive linear functionals (or equivalently, every nonzero weak-*-closed left translation-invariant subspace of L'(G) contains nonzero positive definite functions). By reduction to some special groups Leptin shows that a connected group G satisfies property (L) if and only if G is the direct product of a compact group and a commutative group. In this note we give a short proof of the nontrivial part of that result and of some generalizations to not necessarily connected groups. In particular, if G is almost connected and satisfies property (L), then G is the semidirect product of a vector group V and a compact group K such that K acts on V effectively as a finite group. Let Coo(G) be the continuous functions on G with compact support. For f E Coo(G) let fIH denote its restriction onto the subgroup H and let ,J(y) = f(xy), x, y E G. = fG rp(x)f(x) dx is the canonical bilinear form on L'(G) x L'(G), where dx denotes a fixed left Haar measure on G. Character always means a continuous homomorphism into the circle. THEOREM 1. Let G be a locally compact group. Suppose that every closed proper left ideal in L '(G) is annihilated by nonzero positive linear functionals. Then characters of closed subgroups of G can always be extended as continuous positive definite functions of G. PROOF. Let -y be a character of a closed subgroup H of G and let IY be the right ideal in Coo(G) defined by I = {v E Coo(G); = 0 for all u E Coo(G)}. Received by the editors January 18, 1980. AMS (MOS) subject classifications (1970). Primary 43A15, 43A35, 22D05.

On one-sided harmonic analysis

It is shown that, in locally compact groups satisfying the onesided version of the Wiener property introduced by Leptin, characters of closed subgroups have always continuous positive definite extensions onto the whole group. We give a quick proof that for such groups the connected component of the identity is the direct product of a compact group and a vector group.

A class of symmetric and a class of Wiener group algebras

It is shown that connected groups of polynomial growth and compact extensions of nilpotent group have symmetric group algebras and that the group algebras of discrete solvable groups have the Wiener property.