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.

On rate of decreasing of extremal function in Carleman class

On rate of decreasing of extremal function in Carleman class

Levinson-type theorem and Dyn'kin problems

Questions relating to theorems of Levinson-Sjöberg-Wolf type in complex and harmonic analysis are explored. The well-known Dyn'kin problem of effective estimation of the growth majorant of an analytic function in a neighbourhood of its set of singularities is discussed, together with the problem, dual to it in certain sense, on the rate of convergence to zero of the extremal function in a nonquasianalytic Carleman class in a neighbourhood of a point at which all the derivatives of functions in this class vanish. The first problem was solved by Matsaev and Sodin. Here the second Dyn'kin problem, going back to Bang, is fully solved. As an application, a sharp asymptotic estimate is given for the distance between the imaginary exponentials and the algebraic polynomials in a weighted space of continuous functions on the real line. Bibliography: 24 titles.

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.

Representing Exponential Systems in Spaces of Analytic Functions

This paper is devoted to representing exponential systems in various subspaces of the space H(D) of functions that are analytic in a bounded convex domain D. We consider two kinds of such subspaces: uniformly weighted spaces H(D,𝜑) and spaces of the type of Carleman classes H(D,ℳ).

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.

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.

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.

Power substitution in quasianalytic Carleman classes

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.

Quasianalytic Functional Classes in Jordan Domains of the Complex Plane

In this paper, we study the Carleman classes in Jordan domains of the complex plane. We obtain a quasianalyticity criterion for the regular Carleman classes, which is universal for all weakly uniform domains. The proof is based on the solution of the Dirichlet problem with an unbounded boundary function and a Beurling result on the estimate of the harmonic measure.

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.

A universal criterion for quasi-analytic classes in Jordan domains

Carleman classes in Jordan domains in the complex plane are investigated. A criterion for regular Carleman classes to be quasi-analytic is established, which is universal in a certain sense for all weakly uniform domains. The proof is based on a solution of the Dirichlet problem with unbounded boundary function, and a result on bounds for the harmonic measure due to Beurling plays a substantial role. Bibliography: 20 titles.

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.

Extremal problems in nonquasianalytic Carleman classes. Applications

An extremal problem is considered in the family of functions in a nonquasianalytic Carleman class on a closed interval that vanish together with all derivatives at a point in this interval. Applications to approximation theory and, in particular, to a system of exponentials with exponents satisfying the Fejér (or Levinson) condition are indicated; an asymptotic estimate as is obtained for the distance in between a fixed exponential and the closure of the linear span of other elements of this system. Bibliography: 25 titles.