Consider an equation of the form $f(x)=g(x^k)$, where $k>1$ and $f(x)$ is a function in a given Carleman class of smooth functions. For each $k$, we construct a Carleman-type class which contains all the smooth solutions $g(x)$ to such equations. We prove, under regularity assumptions, that if the original Carleman class is quasianalytic, then so is the new class. The results admit an extension to multivariate functions.
For quasianalytic Denjoy--Carleman differentiable function classes $C^Q$ where the weight sequence $Q=(Q_k)$ is log-convex, stable under derivations, of moderate growth and also an $\mathcal L$-intersection (see 1.6), we prove the following: The category of $C^Q$-mappings is cartesian closed in the sense that $C^Q(E,C^Q(F,G))\cong C^Q(E\times F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^Q$-diffeomorphisms is a regular $C^Q$-Lie group but not better.
For Denjoy--Carleman differential function classes $C^M$ where the weight sequence $M=(M_k)$ is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is $C^M$ if it maps $C^M$-curves to $C^M$-curves. The category of $C^M$-mappings is cartesian closed in the sense that $C^M(E,C^M(F,G))\cong C^M(E\x F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^M$-diffeomorphisms is a $C^M$-Lie group but not better.
Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for |f| outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions. In the first part (this arXiv:math.CA/0208233), we considered quasianaly…
Consider a class of functions of one real variable with the following uniqueness property: if a function f(x) from the class vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for f(x) outside a small exceptional set. Such estimates are well-known and useful for polynomials and analytic functions. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions.
We prove two main results on Denjoy-Carleman classes: (1) a composite function theorem which asserts that a function f(x) in a quasianalytic Denjoy-Carleman class Q, which is formally composite with a generically submersive mapping y=h(x) of class Q, at a single given point in the source (or in the target) of h, can be written locally as f(x) = g(h(x)), where g(y) belongs to a shifted Denjoy-Carleman class Q' ; (2) a statement on a similar loss of regularity for functions definable in the o-minimal structure given by expansion of the real field by restricted functions of quasianalytic class Q.…
In this paper, we explain a sharp phase transition phenomenon which occurs for $L^p$-Carleman classes with exponents $0<p<1$. In principle, these classes are defined as usual, only the traditional $L^\infty$-bounds are replaced by corresponding $L^p$-bounds. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the $L^p$-Carleman class. A particular degenerate instance is when we obtain the $L^p$-Sobolev spaces, analyzed previously by Peetre, following an initial insight by Douady. Peetre found that these $L^p$-Sobol…
Quasianalytic classes are classes of infinitely differentiable functions that satisfy the analytic continuation property enjoyed by analytic functions. Two general examples are quasianalytic Denjoy-Carleman classes (of origin in the analysis of linear partial differential equations) and the class of infinitely differentiable functions that are definable in a polynomially bounded o-minimal structure (of origin in model theory). We prove a generalization to quasianalytic functions of Malgrange's celebrated theorem on the division of infinitely differentiable by real-analytic functions.
This expository article is devoted to the local theory of ultradifferentiable classes of functions, with a special emphasis on the quasianalytic case. Although quasianalytic classes are well-known in harmonic analysis since several decades, their study from the viewpoint of differential analysis and analytic geometry has begun much more recently and, to some extent, has earned them a new interest. Therefore, we focus on contemporary questions closely related to topics in local algebra. We study, in particular, Weierstrass division problems and the role of hyperbolicity, together with propertie…
Let $\mathcal{E}_1(M)^+$ be the local ring of germs at 0 of functions belonging to a given Denjoy-Carleman quasianalytic class in a neighborhood of 0 in $[0,+\infty[$. We show that the ring $\mathcal{E}_1(M)^+$ contains elements that cannot be extended quasianalytically in a neighborhood of 0 in $\mathbb{R}$, unless it coincides with the ring of real-analytic germs.
The classical notion of Łojasiewicz ideals of smooth functions is studied in the context of non-quasianalytic Denjoy-Carleman classes. In the case of principal ideals, we obtain a characterization of Łojasiewicz ideals in terms of properties of a generator. This characterization involves a certain type of estimates that differ from the usual Łojasiewicz inequality. We then show that basic properties of Łojasiewicz ideals in the $\mathcal{C}^\infty$ case have a Denjoy-Carleman counterpart.
In the first part of this work, we consider a polynomial $ ϕ(x,y)=y^d+a_1(x)y^{d-1}+...+a_d(x) $ whose coefficients $ a_j $ belong to a Denjoy-Carleman quasianalytic local ring $ \mathcal{E}_1(M) $. Assuming that $ \mathcal{E}_1(M) $ is stable under derivation, we show that if $ h $ is a germ of $ C^\infty $ function such that $ ϕ(x,h(x))=0 $, then $ h $ belongs to $ \mathcal{E}_1(M) $. This extends a well-known fact about real-analytic functions. We also show that the result fails in general for non-quasianalytic ultradifferentiable local rings. In the second part of the paper, we study a sim…
The article develops techniques for solving equations G(x,y)=0, where G(x,y)=G(x_1,...,x_n,y) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy-Carleman class, or the class of infinitely differentiable functions definable in a polynomially-bounded o-minimal structure). We show that, if G(x,y)=0 has a formal power series solution y=H(x) at some point a, then H is the Taylor expansion at a of a quasianalytic solution y=h(x), where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions…
Let $\mathcal C^M$ denote a Denjoy-Carleman class of $\mathcal C^\infty$ functions (for a given logarithmically-convex sequence $M = (M_n)$). We construct: (1) a function in $\mathcal C^M((-1,1))$ which is nowhere in any smaller class; (2) a function on $\mathbb R$ which is formally $\mathcal C^M$ at every point, but not in $\mathcal C^M(\mathbb R)$; (3) (under the assumption of quasianalyticity) a smooth function on $\mathbb R^p$ ($p \geq 2$) which is $\mathcal C^M$ on every $\mathcal C^M$ curve, but not in $\mathcal C^M(\mathbb R^p)$.
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.
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.
We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$ of $C^\infty$-functions. We require that $\mathcal C$ includes polynomial but excludes flat functions (quasianalyticity) and is closed under composition, derivation, division by a coordinate, and taking the inverse. Examples are quasianalytic Denjoy--Carleman classes, in particular, the class of real analytic functions $C^ω$. We show that there exists a local…
We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy-Carleman classes $\mathcal{C}_M$. For suitable divisors $P(x,t)=x^d+a_1(t)x^{d-1}+\cdots+a_d(t)$ with real-analytic coefficients $a_j$, we show that the quotient and the remainder can be chosen of class $\mathcal{C}_{M^σ}$, where $M^σ=((M_j)^σ)_{j\geq 0}$ and $σ$ is a certain Łojasiewicz exponent $σ$ related to the geometry of the roots of $P$ and verifying $1\leq σ\leq d$. We provide various examples for which $σ$ is optimal, in particular strictly less than $d$, which sharpens earlier results of…
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, i.e., $f\circ c$ is smooth for every smooth curve $c : \mathbb R \to U$. 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 t…
