Spaces Of Ultradistributions Research Articles

An Introduction to Extended Gevrey Regularity

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in Gevrey settings. In this paper, we consider a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview of extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultra distributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic.

Quasianalytic Functionals and Ultradistributions as Boundary Values of Harmonic Functions

We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to Hörmander’s support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions.

Wiener amalgam spaces of quasianalytic ultradistributions

We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global components are either weighted Lp or weighted C0 spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces.

Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin-Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand-Shilov spaces and their dual ultradistribution spaces.

Boundary values of zero solutions of hypoelliptic differential operators in ultradistribution spaces

We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator P(D) = P(D_x, D_t) on {mathbb {R}}^{d+1}. Our work unifies and considerably extends various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. We also give new proofs of several results of Langenbruch (Manuscripta Math. 26:17–35, 1978/79) about distributional boundary values of zero solutions of P(D).

Convolutors of translation-modulation invariant Banach spaces of ultradistributions

We study the space of tempered ultradistributions whose convolutions with test functions are all contained in a given translation-modulation invariant Banach space of ultradistributions. Our main result will be the first structural theorem for the aforementioned space. As an application we consider several extensions of convolution.

Paley–Wiener–Schwartz Type Theorem for Ultradistributional Wavelet Transform

In this paper, we discuss the wavelet transform on the space of ultradistributions of compact support. The Paley–Wiener–Schwartz type theorem for the wavelet transform of both test function and ultradistribution are obtained.

Boundary Values in Ultradistribution Spaces Related to Extended Gevrey Regularity

Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, known as stability under ultradifferential operators in the classical ultradistribution theory, is replaced by a weaker condition, in which the growth properties are controlled by an additional parameter. For that reason, new techniques were used in the proofs. As an application, we discuss the corresponding wave front sets.

On the space of ultradistributions vanishing at infinity

We study the structural and linear topological properties of the space \(\dot{\mathcal {B}}^{\prime *}_{\omega }\) of ultradistributions vanishing at infinity (with respect to a weight function \(\omega \)). Particularly, we show the first structure theorem for \(\dot{\mathcal {B}}^{\prime *}_{\omega }\) under weaker hypotheses than were known so far. As an application, we determine the structure of the S-asymptotic behavior of ultradistributions.

A multidimensional Tauberian theorem for Laplace transforms of ultradistributions

ABSTRACTWe obtain a multidimensional Tauberian theorem for Laplace transforms of Gelfand-Shilov ultradistributions. The result is derived from a Laplace transform characterization of bounded sets in spaces of ultradistributions with supports in a convex acute cone of , also established here.

Very weak solutions to hypoelliptic wave equations

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of “very weak solutions” to the Cauchy problem, that was already successfully used in similar contexts in [12] and [20]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p-evolution equations for higher order operators on Rn or on groups, the results already being new in all these cases.

Local and global trace formulae for smooth hyperbolic diffeomorphisms

We define and study local and global trace formulae for discrete-time uniformly hyperbolic weighted dynamics. We explain first why dynamical determinants are particularly convenient tools to tackle this question. Then we construct counter-examples that highlight that the situation is much less well behaved for smooth dynamics than for real analytic ones. This suggests to study this question for Gevrey dynamics. We do so by constructing an anisotropic space of ultradistributions on which a transfer operator acts as a trace class operator. From this construction, we deduce trace formulae for Gevrey dynamics, as well as bounds on the growth of their dynamical determinants and the asymptotics of their Ruelle resonances.

The continuous wavelet transform on ultradistribution spaces

The continuous wavelet transform on ultradistribution spaces

Wave front sets with respect to Banach spaces of ultradistributions. Characterisation via the short-time fourier transform

We define ultradistributional wave front sets with respect to translation-modulation invariant Banach spaces of ultradistributions having solid Fourier image. The main result is their characterisation by the short-time Fourier transform.

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations

In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our paper [Trans. Amer. Math. Soc. 368 (2016), pp.8481-8498]. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mappings between these spaces on compact manifolds.

Translation–Modulation Invariant Banach Spaces of Ultradistributions

We introduce and study a new class of translation-modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener-Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions $\mathcal{M}^F$ on $\mathbb{R}^{d}$, associated to translation-modulation invariant Banach spaces of ultradistributions $F$ on $\mathbb{R}^{2d}$.

On quasianalytic classes of Gelfand–Shilov type. Parametrix and convolution

We develop a convolution theory for quasianalytic ultradistributions of Gelfand–Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.

Optimal embeddings of ultradistributions into differential algebras

We construct embeddings of spaces of non-quasianalytic ultradistributions into differential algebras enjoying optimal properties in view of a Schwartz type impossibility result, also shown in this article. We develop microlocal analysis in theses algebras consistent with the microlocal analysis in the corresponding spaces of ultradistributions.

On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type $\mathcal{S}^{(p!)}_{(M_p)}$ and Roumieu type $\mathcal{S}^{\{p!\}}_{\{M_p\}}$ are non-trivial if and only if $$ \sup_{p \geq 2}\frac{(\log p)^p}{h^pM_p} < \infty, $$ for all $h > 0$ and some $h > 0$, respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.

G-type spaces of ultradistributions over $$\mathbb {R}^d_+$$ R + d and the Weyl pseudo-differential operators with radial symbols

The first part of the paper is devoted to the G-type spaces i.e. the spaces \(G^\alpha _\alpha (\mathbb R^d_+)\), \(\alpha \ge \)1 and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with completely new justification of obtained results. The Laguerre type expansions of the elements in \(G^\alpha _\alpha (\mathbb R^d_+)\), \(\alpha \ge \)1 and their duals characterise these spaces through the exponential and sub-exponential growth of coefficients. We provide the full topological description and by the nuclearity of \(G_\alpha ^\alpha (\mathbb {R}^d_+)\), \(\alpha \ge \)1 the kernel theorem is proved. The second part is devoted to the class of the Weyl operators with radial symbols belonging to the G-type spaces. The continuity properties of this class of pseudo-differential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo-differential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.