arXiv subjects

Armin Rainer

Publications and source records attributed to Armin Rainer.

At least 19 recordsLinked to original sources

arXiv2026

Effective quasianalytic Remez inequalities on tame sets

We establish a Remez inequality for functions in quasianalytic Denjoy-Carleman classes $\mathcal C_M$ on a large family of fat compact sets $K \subseteq \mathbb R^n$ with tame geometry, including all fat compact sets definable in the o-minimal expansion of $\mathbb R$ by restricted $\mathcal C_M$ functions. The inequality generalizes the classical Remez inequality for polynomials and its quasianalytic versions on convex bodies, replacing the polynomial degree by the Bang degree, an integer associated with the weight $M$ and the size of the function. The constants depend explicitly on the Bang degree and the geometry of $K$. We derive a range of quantitative consequences: estimates for the volume of sublevel sets, comparison of $L^p$-norms, effective inequalities of Lojasiewicz, Harnack, and Markov type with explicit constants, and decay estimates for oscillatory integrals.

math.FA↗

arXiv2026

Eigenvalue stability of Hermitian and normal matrices

The ordered eigenvalues define a Lipschitz map on the real vector space of Hermitian $d \times d$ matrices. We prove that this map acts continuously, but not uniformly continuously, by superposition on the Sobolev spaces $W^{1,q}$, for all $1 \le q < \infty$, on bounded open domains. For $q=\infty$, the action is still well-defined and bounded but not continuous. We show that this stability result extends to normal matrices, where the eigenvalues are naturally interpreted as multivalued Sobolev functions in the sense of Almgren. Several applications are given, including the stability of singular values, condition numbers of matrices, surface area of eigenvalue graphs, and compact self-adjoint operators in Hilbert space.

math.FA↗

arXiv2025

On spaces of arc-smooth maps

It is well-known that a function on an open set in $\mathbb R^d$ is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real analytic version) to suitable closed sets, notably, sets with H\"older boundary and fat subanalytic sets satisfying a necessary topological condition. In this paper, we prove that the resulting set-theoretic identities of function spaces are bornological isomorphisms with respect to their natural locally convex topologies. Extending the results to maps with values in convenient vector spaces, we obtain corresponding exponential laws. Additionally, we show analogous results for special ultradifferentiable Braun-Meise-Taylor classes.

math.CA↗

arXiv2024

Continuity of the solution map for hyperbolic polynomials

Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree $d$ with $C^{d-1,1}$ coefficients are locally Lipschitz and the solution map "coefficients-to-roots" is bounded. We prove continuity of this solution map from hyperbolic polynomials of degree $d$ with $C^d$ coefficients to their increasingly ordered roots with respect to the $C^d$ structure on the source space and the Sobolev $W^{1,q}$ structure, for all $1 \le q<\infty$, on the target space. Continuity fails for $q=\infty$. As a consequence, we obtain continuity of the local surface area of the roots as well as local lower semicontinuity of the area of the zero sets of hyperbolic polynomials. We also discuss applications for the eigenvalues of Hermitian matrices and singular values.

math.FA↗

arXiv2024

On the continuity of the solution map for polynomials

In previous work, we proved that the continuous roots of a monic polynomial of degree $d$ whose coefficients depend in a $C^{d-1,1}$ way on real parameters belong to the Sobolev space $W^{1,q}$ for all $1\le q<d/(d-1)$. This is optimal. We obtained uniform bounds that show that the solution map ``coefficients-to-roots'' is bounded with respect to the $C^{d-1,1}$ and the Sobolev $W^{1,q}$ structures on source and target space, respectively. In this paper, we prove that the solution map is continuous, provided that we consider the $C^d$ structure on the space of coefficients. Since there is no canonical choice of an ordered $d$-tuple of the roots, we work in the space of $d$-valued Sobolev functions equipped with a strong notion of convergence. We also interpret the results in the Wasserstein space on the complex plane.

math.FA↗

arXiv2023

Interpolation of derivatives and ultradifferentiable regularity

Interpolation inequalities for $C^m$ functions allow to bound derivatives of intermediate order $0 < j<m$ by bounds for the derivatives of order $0$ and $m$. We review various interpolation inequalities for $L^p$-norms ($1 \le p \le \infty$) in arbitrary finite dimensions. They allow us to study ultradifferentiable regularity by lacunary estimates in a comprehensive way, striving for minimal assumptions on the weights.

math.FA↗

arXiv2023

On real analytic functions on closed subanalytic domains

We show that a function $f : X \to \mathbb R$ defined on a closed uniformly polynomially cuspidal set $X$ in $\mathbb R^n$ is real analytic if and only if $f$ is smooth and all its composites with germs of polynomial curves in $X$ are real analytic. The degree of the polynomial curves needed for this is effectively related to the regularity of the boundary of $X$. For instance, if the boundary of $X$ is locally Lipschitz, then polynomial curves of degree $2$ suffice. In this Lipschitz case, we also prove that a function $f : X \to \mathbb R$ is real analytic if and only if all its composites with germs of quadratic polynomial maps in two variables with images in $X$ are real analytic; here it is not necessary to assume that $f$ is smooth.

math.CA↗

arXiv2023

Perturbation theory of polynomials and linear operators

This survey revolves around the question how the roots of a monic polynomial (resp. the spectral decomposition of a linear operator), whose coefficients depend in a smooth way on parameters, depend on those parameters. The parameter dependence of the polynomials (resp. operators) ranges from real analytic over $C^\infty$ to differentiable of finite order with often drastically different regularity results for the roots (resp. eigenvalues and eigenvectors). Another interesting point is the difference between the perturbation theory of hyperbolic polynomials (where, by definition, all roots are real) and that of general complex polynomials. The subject, which started with Rellich's work in the 1930s, enjoyed sustained interest through time that intensified in the last two decades, bringing some definitive optimal results. Throughout we try to explain the main proof ideas; Rellich's theorem and Bronshtein's theorem on hyperbolic polynomials are presented with full proofs. The survey is written for readers interested in singularity theory but also for those who intend to apply the results in other fields.

math.FA↗

arXiv2023

Definable Lipschitz selections for affine-set valued maps

Whitney's extension problem, i.e., how one can tell whether a function $f : X \to \mathbb R$, $X \subseteq \mathbb R^n$, is the restriction of a $C^m$-function on $\mathbb R^n$, was solved in full generality by Charles Fefferman in 2006. In this paper, we settle the $C^{1,\omega}$-case of a related conjecture: given that $f$ is semialgebraic and $\omega$ is a semialgebraic modulus of continuity, if $f$ is the restriction of a $C^{1,\omega}$-function then it is the restriction of a semialgebraic $C^{1,\omega}$-function. We work in the more general setting of sets that are definable in an o-minimial expansion of the real field. An ingenious argument of Brudnyi and Shvartsman relates the existence of $C^{1,\omega}$-extensions to the existence of Lipschitz selections of certain affine-set valued maps. We show that if a definable affine-set valued map has Lipschitz selections then it also has definable Lipschitz selections. In particular, we obtain a Lipschitz solution (more generally, $\omega$-H\"older solution, for any definable modulus of continuity $\omega$) of the definable Brenner-Epstein-Hochster-Koll\'ar problem. In most of our results we have control over the respective (semi)norms.

math.LO↗

arXiv2023

Uniform extension of definable $C^{m,\omega}$-Whitney jets

We show that definable Whitney jets of class $C^{m,\omega}$, where $m$ is a nonnegative integer and $\omega$ is a modulus of continuity, are the restrictions of definable $C^{m,\omega}$-functions; "definable" refers to an arbitrary given o-minimal expansion of the real field. This is true in a uniform way: any definable bounded family of Whitney jets of class $C^{m,\omega}$ extends to a definable bounded family of $C^{m,\omega}$-functions. We also discuss a uniform $C^m$-version and how the extension depends on the modulus of continuity.

math.LO↗

arXiv2022

Quantitative tame properties of differentiable functions with controlled derivatives

We show that differentiable functions, defined on a convex body $K \subseteq \mathbb R^d$, whose derivatives do not exceed a suitable given sequence of positive real numbers share many properties with polynomials. The role of the degree of a polynomial is hereby played by an integer associated with the given sequence of reals, the diameter of $K$, and a real parameter linked to the $C^0$-norm of the function. We give quantitative information on the size of the zero set, show that it admits a local parameterization by Sobolev functions, and prove an inequality of Remez-type. From the latter, we deduce several consequences, for instance, a bound on the volume of sublevel sets and a comparison of $L^p$-norms reversing H\"older's inequality. The validity of many of the results only depends on the derivatives up to some finite order; the order can be specified in terms of the given data.

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 Fr\'echet spaces, a Phragm\'en-Lindel\"of theorem, and H\"ormander's solution of the $\bar{\partial}$-problem.

math.FA↗

arXiv2022

H\"older--Zygmund classes on smooth curves

We prove that a function in several variables is in the local Zygmund class $\mathcal Z^{m,1}$ if and only if its composite with every smooth curve is of class $\mathcal Z^{m,1}$. This complements the well-known analogous result for local H\"older--Lipschitz classes $\mathcal C^{m,\alpha}$ which we reprove along the way. We demonstrate that these results generalize to mappings between Banach spaces and use them to study the regularity of the superposition operator $f_* : g \mapsto f \circ g$ acting on the global Zygmund space $\Lambda_{m+1}(\mathbb R^d)$. We prove that, for all integers $m,k\ge 1$, the map $f_* : \Lambda_{m+1}(\mathbb R^d) \to \Lambda_{m+1}(\mathbb R^d)$ is of Lipschitz class $\mathcal C^{k-1,1}$ if and only if $f \in \mathcal Z^{m+k,1}(\mathbb R)$.

math.FA↗

arXiv2021

Ultradifferentiable extension theorems: a survey

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney's classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy-Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun-Meise-Taylor classes based on their representation as intersections and unions of Denjoy-Carleman classes.

math.FA↗

arXiv2021

Arc-smooth functions and cuspidality of sets

A function $f$ is arc-smooth if the composite $f\circ c$ with every smooth curve $c$ in its domain of definition is smooth. On open sets in smooth manifolds the arc-smooth functions are precisely the smooth functions by a classical theorem of Boman. Recently, we extended this result to certain tame closed sets (namely, H\"older sets and simple fat subanalytic sets). In this paper we link, in a precise way, the cuspidality of the (boundary of the) set to the loss of regularity, i.e., how many derivatives of $f\circ c$ are needed in order to determine the derivatives of $f$. We also discuss how flatness of $f \circ c$ affects flatness of $f$. Besides H\"older sets and subanalytic sets we treat sets that are definable in certain polynomially bounded o-minimal expansions of the real field.

math.CA↗

arXiv2021

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↗

arXiv2021

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 $\Phi$ instead of the powers, and characterize when $\Phi \circ f \in \mathcal C$ implies $f \in \mathcal C$.

math.CA↗

arXiv2021

Roots of Garding hyperbolic polynomials

We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces.

math.CA↗