arXiv subjects

David Nicolas Nenning

Publications and source records attributed to David Nicolas Nenning.

11 recordsLinked to original sources

arXiv2023

Ultradifferentiable classes of entire functions

We study classes of ultradifferentiable functions defined in terms of small weight sequences violating standard growth and regularity requirements. First, we show that such classes can be viewed as weighted spaces of entire functions for which the crucial weight is given by the associated weight function of the so-called conjugate weight sequence. Moreover, we generalize results from M. Markin from the so-called small Gevrey-setting to arbitrary convenient families of (small) sequences and show how the corresponding ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated to an evolution equation problem). Finally, we study the connection between small sequences and the recent notion of dual sequences introduced in the PhD-thesis of J. Jiménez-Garrido.

math.FA↗

arXiv2023

On generalized definitions of ultradifferentiable classes

We show that the ultradifferentiable-like classes of smooth functions introduced and studied by S. Pilipović, N. Teofanov and F. Tomić are special cases of the general framework of spaces of ultradifferentiable functions defined in terms of weight matrices in the sense of A. Rainer and the third author. We study classes "beyond geometric growth factors" defined in terms of a weight sequence and an exponent sequence, prove that these new types admit a weight matrix representation and transfer known results from the matrix-type to such a non-standard ultradifferentiable setting.

math.FA↗

arXiv2022

The Borel map in the mixed Beurling setting

The Borel map takes a smooth function to its infinite jet of derivatives (at zero). We study the restriction of this map to ultradifferentiable classes of Beurling type in a very general setting which encompasses the classical Denjoy-Carleman and Braun-Meise-Taylor classes. More precisely, we characterize when the Borel image of one class covers the sequence space of another class in terms of the two weights that define the classes. We present two independent solutions to this problem, one by reduction to the Roumieu case and the other by dualization of the involved Fréchet spaces, a Phragmén-Lindelöf theorem, and Hörmander's solution of the $\bar{\partial}$-problem.

math.FA↗

arXiv2022

On optimal solutions of the Borel problem in the Roumieu case

The Borel problem for Denjoy--Carleman and Braun--Meise--Taylor classes has well-known optimal solutions. The unified treatment of these ultradifferentiable classes by means of one-parameter families of weight sequences allows to compare these optimal solutions. We determine the relations among them and give conditions for their equivalence in the Roumieu case.

math.CV↗

arXiv2022

Nonlinear conditions for ultradifferentiability: a uniform approach

Recent work showed that a theorem of Joris (that a function $f$ is smooth if two coprime powers of $f$ are smooth) is valid in a wide variety of ultradifferentiable classes $\mathcal C$. The core of the proof was essentially $1$-dimensional. In certain cases a multidimensional version resulted from subtle reduction arguments, but general validity, notably in the quasianalytic setting, remained open. In this paper we give a uniform proof which works in all cases and dimensions. It yields the result even on infinite dimensional Banach spaces and convenient vector spaces. We also consider more general nonlinear conditions, namely general analytic germs $Φ$ instead of the powers, and characterize when $Φ\circ f \in \mathcal C$ implies $f \in \mathcal C$.

math.CA↗

arXiv2021

Ultraholomorphic sectorial extensions of Beurling type

We prove sectorial extension theorems for ultraholomorphic function classes of Beurling type defined by weight functions with a controlled loss of regularity. The proofs are based on a reduction lemma, due to the second author, which allows to extract the Beurling from the Roumieu case, which was treated recently by Jiménez-Garrido, Sanz, and the third author. In order to have control on the opening of the sectors, where the extensions exist, we use the (mixed) growth index and the order of quasianalyticity of weight functions. As a consequence we obtain corresponding extension results for classes defined by weight sequences. Additionally, we give information on the existence of continuous linear extension operators.

math.FA↗

arXiv2021

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.

math.CA↗

arXiv2019

Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the $\bar \partial$-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensions and of the FBI transform. This allows to extend its definition to ultradifferentiable manifolds. We also discuss ultradifferentiable versions of the elliptic regularity theorem and obtain a general quasianalytic Holmgren uniqueness theorem.

math.AP↗

arXiv2019

On time-dependent Besov vector fields and the regularity of their flows

We show ODE-closedness for a large class of Besov spaces $\mathcal{B}^{n,α,p}(\mathbb{R}^d,\mathbb{R}^d)$, where $n \geq 1,~α\in (0,1],~p \in [1,\infty]$. ODE-closedness means that pointwise time-dependent $\mathcal{B}^{n,α,p}$-vector fields $u$ have unique flows $Φ_u \in \operatorname{Id} + \mathcal{B}^{n,α,p}(\mathbb{R}^d,\mathbb{R}^d)$. The class of vector fields under consideration contains as a special case the class of Bochner integrable vector fields $L^1(I, \mathcal{B}^{n,α,p}(\mathbb{R}^d,\mathbb{R}^d))$. In addition, for $n \geq 2$ and $α< β$, we show continuity of the flow mapping $L^1(I,\mathcal{B}^{n,β,p}(\mathbb{R}^d,\mathbb{R}^d)) \rightarrow C(I,\mathcal{B}^{n,α,p}(\mathbb{R}^d,\mathbb{R}^d)), ~ u \mapsto Φ_u-\operatorname{Id}$. We even get $γ$-Hölder continuity for any $γ< β- α$.

math.CA↗

arXiv2017

The Trouvé group for spaces of test functions

The Trouvé group $\mathcal G_{\mathcal A}$ from image analysis consists of the flows at a fixed time of all time-dependent vectors fields of a given regularity $\mathcal A(\mathbb R^d,\mathbb R^d)$. For a multitude of regularity classes $\mathcal A$, we prove that the Trouvé group $\mathcal G_{\mathcal A}$ coincides with the connected component of the identity of the group of orientation preserving diffeomorphims of $\mathbb R^d$ which differ from the identity by a mapping of class $\mathcal A$. We thus conclude that $\mathcal G_{\mathcal A}$ has a natural regular Lie group structure. In many cases we show that the mapping which takes a time-dependent vector field to its flow is continuous. As a consequence we obtain that the scale of Bergman spaces on the polystrip with variable width is stable under solving ordinary differential equations.

math.CA↗

arXiv2017

On groups of Hölder diffeomorphisms and their regularity

We study the set $\mathcal D^{n,β}(\mathbb R^d)$ of orientation preserving diffeomorphisms of $\mathbb R^d$ which differ from the identity by a Hölder $C^{n,β}_0$-mapping, where $n \in \mathbb N_{\ge 1}$ and $β\in (0,1]$. We show that $\mathcal D^{n,β}(\mathbb R^d)$ forms a group, but left translations in $\mathcal D^{n,β}(\mathbb R^d)$ are in general discontinuous. The groups $\mathcal D^{n,β-}(\mathbb R^d) := \bigcap_{α< β} \mathcal D^{n,α}(\mathbb R^d)$ (with its natural Fréchet topology) and $\mathcal D^{n,β+}(\mathbb R^d) := \bigcup_{α> β} \mathcal D^{n,α}(\mathbb R^d)$ (with its natural inductive locally convex topology) however are $C^{0,ω}$ Lie groups for any slowly vanishing modulus of continuity $ω$. In particular, $\mathcal D^{n,β-}(\mathbb R^d)$ is a topological group and a so-called half-Lie group (with smooth right translations). We prove that the Hölder spaces $C^{n,β}_0$ are ODE closed, in the sense that pointwise time-dependent $C^{n,β}_0$-vector fields $u$ have unique flows $Φ$ in $\mathcal D^{n,β}(\mathbb R^d)$. This includes, in particular, all Bochner integrable functions $u \in L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d))$. For the latter and $n\ge 2$, we show that the flow map $L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d)) \to C([0,1],\mathcal D^{n,α}(\mathbb R^d))$, $u \mapsto Φ$, is continuous (even $C^{0,β-α}$), for every $α< β$. As an application we prove that the corresponding Trouvé group $\mathcal G_{n,β}(\mathbb R^d)$ from image analysis coincides with the connected component of the identity of $\mathcal D^{n,β}(\mathbb R^d)$.

math.CA↗