Denjoy-Carleman Classes Research Articles

Ellipticity and the problem of iterates in Denjoy–Carleman classes

AbstractIn 1978 Métivier showed that a linear differential operator P with analytic coefficients is elliptic if and only if the theorem of iterates holds for P with respect to any non-analytic Gevrey class. In this paper we extend this theorem to Denjoy–Carleman classes given by strongly non-quasianalytic weight sequences. The proof involves a new way to construct optimal functions in Denjoy–Carleman classes via Fourier integrals, which might be of independent interest. Moreover, we point out that the analogous statement for Braun–Meise–Taylor classes given by weight functions cannot hold. This signifies an important difference in the properties of Denjoy–Carleman classes and Braun–Meise–Taylor classes, respectively.

The Metivier inequality and ultradifferentiable hypoellipticity

AbstractIn 1980, Métivier characterized the analytic (and Gevrey) hypoellipticity of ‐solvable partial linear differential operators by a priori estimates. In this note, we extend this characterization to ultradifferentiable hypoellipticity with respect to Denjoy–Carleman classes given by suitable weight sequences. We also discuss the case when the solutions can be taken as hyperfunctions and present some applications.

An interpolation problem in the Denjoy–Carleman classes

AbstractInspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real‐analytic coefficients, we consider the following question. Given a smooth function defined on and given an increasing divergent sequence of positive integers such that the derivative of order of f has a growth of the type , when can we deduce that f is a function in the Denjoy–Carleman class ? We provide a positive result and show that a suitable condition on the gaps between the terms of the sequence is needed.

Polynomial inequalities, o-minimality and Denjoy–Carleman classes

We study several intimately related problems in the theory of multivariate polynomial inequalities. Firstly, given a map h in certain quasianalytic Denjoy–Carleman classes, we show how to decide whether the image under h of a set satisfying Markov's (resp. Nikolskii's) inequality satisfies Markov's (resp. Nikolskii's) inequality. Our approach relies heavily on the theory of o-minimal structures, particularly on the work of Rolin, Speissegger and Wilkie. Secondly, we establish the relation between Markov's inequality and Nikolskii's inequality (both in general setting and in o-minimal setting). Thirdly, we prove that each compact, definable (in an o-minimal structure) set satisfying Markov's inequality is fat. In particular, this solves in the o-minimal category the well-known problem of nonpluripolarity of sets satisfying Markov's inequality. And lastly, we develop a unified method of polynomial approximation for various classes of smooth functions of several variables.

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.

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

AbstractIn this paper we study microlocal regularity of a ‐solution u of the equation where is ultradifferentiable in the variables and holomorphic in the variables . We proved that if is a regular Denjoy–Carleman class (including the quasianalytic case) then: where is the Denjoy–Carleman wave‐front set of u and is the characteristic set of the linearized operator :

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.

Approximations in globally subanalytic and Denjoy-Carleman classes

Efroymson's Approximation Theorem asserts that if f is a C0 semialgebraic mapping on a C∞ semialgebraic submanifold M of Rn and if ε:M→R is a positive continuous semialgebraic function then there is a C∞ semialgebraic function g:M→R such that |f−g|<ε. We prove a generalization of this result to the globally subanalytic category. Our theorem actually holds in a larger framework since it applies to every function which is definable in a polynomially bounded o-minimal structure (expanding the real field) that admits C∞ cell decomposition. We also establish approximation theorems for Lipschitz and C1 definable functions.

Sharp Estimates for Blowing Down Functions in a Denjoy–Carleman Class

Abstract If $F$ is a $C^{\infty }$ function whose composition $F \circ \sigma $ with a blowing-up ${\sigma }$ belongs to a Denjoy–Carleman class $C_M$, then $F$, in general, belongs to a larger class $C_{M^{(2)}}$; that is, there is a loss of regularity. We show that this loss of regularity is sharp. In particular, the loss of regularity of Denjoy–Carleman classes is intrinsic to arguments involving resolution of singularities.

Fourier analysis for Denjoy–Carleman classes on the torus

In the present article, we develop Fourier series for a family of classes of Romieu type of ultradifferentiable functions and ultradistributions on the torus, usually known as Denjoy-Carleman classes. In this setting, as applications, we extend the Greenfield-Wallach Theorem and, through a conjugation, we characterize global hypoellipticity for a class of systems of real vector fields of tube type.

Factorization in Denjoy-Carleman classes associated to representations of [formula omitted

For two types of moderate growth representations of (Rd,+) on sequentially complete locally convex Hausdorff spaces (including F-representations [14]), we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [14, Conjecture 6.4] for analytic vectors of representations of G=(Rd,+). As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.

Transfer operators for ultradifferentiable expanding maps of the circle

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$, providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$.

Functions with Ultradifferentiable Powers

We study the regularity of smooth functions f defined on an open subset of $${\mathbb {R}}^n$$ and such that, for certain integers $$p\ge 2$$ , the powers $$f^p :x\mapsto (f(x))^p$$ belong to a Denjoy–Carleman class $${\mathcal {C}}_M$$ associated with a suitable weight sequence M. Our main result is a statement analogous to a classic theorem of H. Joris on $${\mathcal {C}}^\infty $$ functions: if a function $$f:{\mathbb {R}}\rightarrow {\mathbb {C}}$$ is such that both functions $$f^p$$ and $$f^q$$ with $$\gcd (p,q)=1$$ are of class $${\mathcal {C}}_M$$ on $${\mathbb {R}}$$ , and if the weight sequence M satisfies a standard condition known as moderate growth, then f itself is of class $${\mathcal {C}}_M$$ . It is also shown that the result is no longer true without the moderate growth assumption. Various ancillary results, corollaries and examples are presented.

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.

MICROLOCAL REGULARITY FOR MIZOHATA TYPE DIFFERENTIAL OPERATORS

We investigate interesting connections between Mizohata type vector fields and microlocal regularity of nonlinear first-order PDEs, establishing results in Denjoy–Carleman classes and real analyticity results in the linear case.

Solutions of quasianalytic equations

The article develops techniques for solving equations $$G(x,y)=0$$ , where $$G(x,y)=G(x_1,\ldots ,x_n,y)$$ is a function in a given quasianalytic class (for example, a quasianalytic Denjoy–Carleman class, or the class of $${\mathcal C}^\infty $$ 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, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed.

Malgrange division by quasianalytic functions

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.

Estimates for Weierstrass division in ultradifferentiable classes

We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy–Carleman classes CM. For suitable divisors P(x,t)=xd+a1(t)xd−1+⋯+ad(t) with real-analytic coefficients aj, we show that the quotient and the remainder can be chosen of class CMσ, where Mσ=((Mj)σ)j≥0 and σ is a certain Łojasiewicz exponent related to the geometry of the roots of P and verifying 1≤σ≤d. We provide various examples for which σ is optimal, in particular strictly less than d, which sharpens earlier results of Bronshtein and of Chaumat–Chollet.

Pathological Phenomena in Denjoy–Carleman Classes

AbstractLet M denote a Denjoy–Carleman class of ∞ functions (for a given logarithmically-convex sequence M = (Mn)). We construct: (1) a function in M((−1, 1)) that is nowhere in any smaller class; (2) a function on ℝ that is formally M at every point, but not in M (ℝ); (3) (under the assumption of quasianalyticity) a smooth function on ℝp (p ≥ 2) that is M on every M curve, but not inM (ℝp).

Quantifier elimination in quasianalytic structures via non-standard analysis

The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here. We also provide a quasianalytic exposition of the results concerning rectilinearization of terms and of definable functions from our earlier research. It will be used to demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented merely by the reciprocal function. More precisely, we construct a plane definable curve, which indicates both that the classical theorem by J. Denef and L. van den Dries as well as \L{}ojasiewicz's theorem that every subanalytic curve is semianalytic are no longer true for quasianalytic structures. Besides rectilinearization of terms, our construction makes use of some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces. Note finally that this paper comprises our earlier preprints on the subject from May 2012.