Roumieu Type Research Articles

Stability properties of ultraholomorphic classes of Roumieu-type defined by weight matrices

We characterize several stability properties, such as inverse or composition closedness, for ultraholomorphic function classes of Roumieu type defined in terms of a weight matrix. In this way we transfer and extend known results from J. Siddiqi and M. Ider, from the weight sequence setting and in sectors not wider than a half-plane, to the weight matrix framework and for sectors in the Riemann surface of the logarithm with arbitrary opening. The key argument rests on the construction, under suitable hypotheses, of characteristic functions in these classes for unrestricted sectors. As a by-product, we obtain new stability results when the growth control in these classes is expressed in terms of a weight sequence, or of a weight function in the sense of Braun–Meise–Taylor.

A note on the barrelledness of weighted PLB-spaces of ultradifferentiable functions

In this note, we consider weighted PLB-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work (2022). We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main result of our above mentioned work. In particular, we obtain that the multiplier space of the Gelfand–Shilov space \Sigma^r_s(\mathbb{R}^d) of Beurling type is ultrabornological, whereas the one of the Gelfand–Shilov space \mathcal{S}^r_s(\mathbb{R}^d) of Roumieu type is not barrelled.

Nonlinear Conditions for Ultradifferentiability

A remarkable theorem of Joris states that a function f is C^infty if two relatively prime powers of f are C^infty . Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.

Ultraholomorphic extension theorems in the mixed setting

The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.

Global hypoelliptic vector fields in ultradifferentiable classes and normal forms

In this paper we prove that a global ω-hypoelliptic vector field on the torus Tn can be transformed by a Eω diffeomorphism of Tn into a vector field with constant coefficients which satisfy a Diophantine condition in terms of the weight function ω. Thereby, we extend previous work by Chen and Chi to a bigger scale of spaces, namely, in the setting of ultradifferentiable classes and ultradistributions of Beurling and Roumieu type.

The spaces of formal power series of class M of Roumieu type and of Beurling type

In this article, we introduce the space of formal power series of class $\mathbb M$ of Roumieu (resp., Beurling) type, which is the generalization of the space of entire functions of normal (resp., minimal) type with respect to a proximate order. And we characterize continuous endomorphisms of these spaces.

A linear topological invariant for spaces of quasianalytic functions of Roumieu type

We show that the spaces E{ω}(Ω) of ultradifferentiable functions of Roumieu type satisfy the dual interpolation estimate for small theta, where ω is a quasianalytic weight function and Ω is an arbitrary open subset of Rd. This result was previously shown by Bonet and Domański [2] under the additional assumptions that Ω is convex and ω satisfies the condition (α1). In particular, our work solves Problem 9.7 in [1].

Solution to the Stieltjes moment problem in Gelfand–Shilov spaces

We characterize the surjectivity and the existence of a continuous linear right inverse of the Stieltjes moment mapping on Gelfand-Shilov spaces, both of Beurling and Roumieu type, in terms of their defining weight sequence. As a corollary, we obtain some new results about the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane.

Arc-smooth functions on closed sets

By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb{R}^{d}$ is smooth (${\mathcal{C}}^{\infty }$) if and only if it is arc-smooth, that is, $f\,\circ \,c$ is smooth for every smooth curve $c:\mathbb{R}\rightarrow U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman’s theorem on fat closed sets with Hölder boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If $X\subseteq \mathbb{R}^{d}$ is any such set and $f:X\rightarrow \mathbb{R}$ is arc-smooth, then $f$ extends to a smooth function defined on $\mathbb{R}^{d}$. We also get a version of the Bochnak–Siciak theorem on all closed fat subanalytic sets and all closed sets with Hölder boundary: if $f:X\rightarrow \mathbb{R}$ is the restriction of a smooth function on $\mathbb{R}^{d}$ which is real analytic along all real analytic curves in $X$, then $f$ extends to a holomorphic function on a neighborhood of $X$ in $\mathbb{C}^{d}$. Similar results hold for non-quasianalytic Denjoy–Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.

Dynamics of weighted composition operators on function spaces defined by local properties

We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of $\mathbb{R}^d$, as well as results characterizing topological transitiviy of such operators on the space of real analytic functions on an open subset of $\mathbb{R}^d$.

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.

Solvability of the inhomogeneous Cauchy–Riemann equation in projective weighted spaces

We establish an analog of Hormander’s Theorem on solvability of the inhomogeneous Cauchy–Riemann equation for a space of measurable functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence defining the space. The same conditions guarantee the weak reducibility of the corresponding space of entire functions. Basing on these results, we solve the problem of describing the multipliers in weighted spaces of entire functions with the projective and inductive-projective topological structure. Applications are obtained to convolution operators in the spaces of ultradifferentiable functions of Roumieu type.

Semilinear pseudodifferential equations in spaces of tempered ultradistributions

We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is a pseudodifferential operator of infinite order satisfying a suitable ellipticity condition we prove a regularity result in the functional setting above for weak Sobolev type solutions.

A sequential approach to ultradistribution spaces

We introduce and investigate two types of the space U* of s-ultradistributions meant as equivalence classes of suitably defined fundamental sequences of smooth functions; we prove the existence of an isomorphism between U* and the respective space D?* of ultradistributions: of Beurling type if * = (p!t) and of Roumieu type if * = {p!t}. We also study the spaces T * and ?T * of t-ultradistributions and ?t-ultradistributions, respectively, and show that these spaces are isomorphic with the space S?* of tempered ultradistributions both in the Beurling and the Roumieu case.

The convenient setting for Denjoy–Carleman differentiable mappings of Beurling and Roumieu type

We prove in a uniform way that all Denjoy–Carleman differentiable function classes of Beurling type $$C^{(M)}$$ and of Roumieu type $$C^{\{M\}}$$ , admit a convenient setting if the weight sequence $$M=(M_k)$$ is log-convex and of moderate growth: For $$\mathcal C$$ denoting either $$C^{(M)}$$ or $$C^{\{M\}}$$ , the category of $$\mathcal C$$ -mappings is cartesian closed in the sense that $$\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\times F, G)$$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $$\mathcal C$$ -diffeomorphisms is a regular $$\mathcal C$$ -Lie group if $$\mathcal C \supseteq C^\omega $$ , but not better.

An exotic zoo of diffeomorphism groups on $$\mathbb {R}^n$$ R n

Let $C^{[M]}$ be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence $M=(M_k)$ is log-convex and has moderate growth. We prove that the groups ${\operatorname{Diff}}\mathcal{B}^{[M]}(\mathbb{R}^n)$, ${\operatorname{Diff}}W^{[M],p}(\mathbb{R}^n)$, ${\operatorname{Diff}}{\mathcal{S}}{}_{[L]}^{[M]}(\mathbb{R}^n)$, and ${\operatorname{Diff}}\mathcal{D}^{[M]}(\mathbb{R}^n)$ of $C^{[M]}$-diffeomorphisms on $\mathbb{R}^n$ which differ from the identity by a mapping in $\mathcal{B}^{[M]}$ (global Denjoy--Carleman), $W^{[M],p}$ (Sobolev-Denjoy-Carleman), ${\mathcal{S}}{}_{[L]}^{[M]}$ (Gelfand--Shilov), or $\mathcal{D}^{[M]}$ (Denjoy-Carleman with compact support) are $C^{[M]}$-regular Lie groups. As an application we use the $R$-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes $W^{[M],1}$, ${\mathcal{S}}{}_{[L]}^{[M]}$, and $\mathcal{D}^{[M]}$. Here we find some surprising groups with continuous left translations and $C^{[M]}$ right translations (called half-Lie groups), which, however, also admit $R$-transforms.

Parametrices and hypoellipticity for pseudodifferential operators on spaces of tempered ultradistributions

We construct parametrices for a class of pseudodifferential operators of infinite order acting on spaces of tempered ultradistributions of Beurling and Roumieu type. As a consequence we obtain a result of hypoellipticity in these spaces.

Anti-Wick and Weyl quantization on ultradistribution spaces

The connection between the Anti-Wick and the Weyl quantization is given for certain class of global symbols, for which corresponding pseudodifferential operators act continuously on the space of tempered ultradistributions of Beurling, respectively of Roumieu type. The largest subspace of ultradistributions is found for which the convolution with the Gaussian kernel exists. This gives a way to extend the definition of Anti-Wick quantization for symbols that are not necessarily tempered ultradistributions.

Generic results in classes of ultradifferentiable functions

Let E be a Denjoy–Carleman class of ultradifferentiable functions of Beurling type on the real line that strictly contains another class F of Roumieu type. We show that the set S of functions in E that are nowhere in the class F is large in the topological sense (it is residual), in the measure theoretic sense (it is prevalent), and that S∪{0} contains an infinite dimensional linear subspace (it is lineable). Consequences for the Gevrey classes are given. Similar results are also obtained for classes of ultradifferentiable functions defined imposing conditions on the Fourier–Laplace transform of the function.

Pseudodifferential operators of infinite order in spaces of tempered ultradistributions

Specific global symbol classes and corresponding pseudodifferential operators of infinite order that act continuously on the space of tempered ultradistributions of Beurling and Roumieu type are constructed. For these classes, symbolic calculus is developed.