arXiv subjects

Vincent Thilliez

Publications and source records attributed to Vincent Thilliez.

9 recordsLinked to original sources

arXiv2019

Functions with ultradifferentiable powers

We study the regularity of smooth functions $f$ defined on an open set of $\mathbb{R}^n$ and such that, for certain integers $p\geq 2$, 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 $f:\mathbb{R}\to\mathbb{R}$ 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 the so-called moderate growth assumption, then $f$ itself is of class $\mathcal{C}_M$. Various ancillary results, corollaries and examples are presented.

math.CA↗

arXiv2016

Estimates for Weierstrass division in ultradifferentiable classes

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 Bronshtein and of Chaumat-Chollet.

math.CV↗

arXiv2013

Łojasiewicz ideals in Denjoy-Carleman classes

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.

math.CA↗

arXiv2010

On the non-extendability of quasianalytic germs

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.

math.CA↗

arXiv2008

Smooth solutions of quasianalytic or ultraholomorphic equations

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 similar problem in the framework of ultraholomorphic functions on sectors of the Riemann surface of the logarithm. We obtain a result that includes suitable non-quasianalytic situations.

math.CA↗

arXiv2008

On quasianalytic local rings

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 properties of ideals of quasianalytic germs. Incidentally, we also present a simplified proof of Carleman's theorem on the non-surjectivity of the Borel map in the quasianalytic case.

math.CA↗

arXiv2006

Division by Flat Ultradifferentiable Functions and Sectorial Extensions

We consider classes $ \mathcal{A}_M(S) $ of functions holomorphic in an open plane sector $ S $ and belonging to a strongly non-quasianalytic class on the closure of $ S $. In $ \mathcal{A}_M(S) $, we construct functions which are flat at the vertex of $ S $ with a sharp rate of vanishing. This allows us to obtain a Borel-Ritt type theorem for $ \mathcal{A}_M(S) $ extending previous results by Schmets and Valdivia. We also derive a division property for ideals of flat ultradifferentiable functions, in the spirit of a classical $ C^\infty $ result of Tougeron.

math.CA↗

arXiv2006

On the Stability of Analytic Germs under Ultradifferentiable Perturbations

Let $ f$ be a real-analytic function germ whose critical locus contains a given real-analytic set $ X $, and let $ Y $ be a germ of closed subset of $ \mathbb{R}^n $ at the origin. We study the stability of $ f $ under perturbations $ u $ that are flat on $ Y $ and that belong to a given Denjoy-Carleman non-quasianalytic class. We obtain a condition ensuring that $ f+u=f\circΦ$ where $ Φ$ is a germ of diffeomorphism whose components belong to a (generally larger) Denjoy-Carleman class. Roughly speaking, this condition involves a Łojasiewicz-type separation property between $ Y $ and the complex zeros of a certain ideal associated with $ f $ and $ X $. The relationship between the Denjoy-Carleman classes of $ u$ and $ Φ$ is controlled precisely by the inequality. This result extends, and simplifies, former work of the author on germs with isolated critical points.

math.CA↗

arXiv2004

Infinite determinacy on a closed set for smooth germs with non-isolated singularities

We give necessary and sufficient conditions of infinite determinacy for smooth function germs whose critical locus contains a given set. This set is assumed to be the zero variety X of some analytic map germ having maximal rank on a dense subset of X. We obtain a result in terms of Lojasiewicz estimates which extends, in particular, previous works by Sun & Wilson on line singularities, and by Grandjean on singularities of codimension 1 or 2.

math.CA↗