相关论文: {\L}ojasiewicz ideals in Denjoy-Carleman classes
Let $\Omega \subset \mathbb{R}^n $ be an open set, and let $\mathcal{E}(\Omega)$ be the ring of infinitely differentiable functions on $\Omega$. For an ideal $I \subset \mathcal{E}(\Omega)$, we denote by $Z(I)$ its zero set. A classical…
For an ideal of smooth functions that is either {\L}ojasiewicz or weakly {\L}ojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety in terms of the global {\L}ojasiewicz radical and Whitney…
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)…
This paper is devoted to proving the general {\L}ojasiewicz inequality, in both the definable and subanalytic cases, under the most relaxed assumptions. It means that we drop the usual continuity and compactness assumptions. In the second…
The equivalence of the Kohn finite ideal type and the D'Angelo finite type with the subellipticity of the $\bar\partial$-Neumann problem is extended to pseudoconvex domains in $C^n$ whose defining function is in a Denjoy-Carleman…
We give an expression for the {\L}ojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the {\L}ojasiewicz exponent of the gradient…
This expository article is devoted to the notion of quasianalytic classes and the Borel mapping. Although quasianalytic classes are well known in analysis since several decades. We are interested in certain properties of Denjoy-Carleman's…
We show that a version of the desingularization theorem of Hironaka holds for certain classes of infinitely differentiable functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the…
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…
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…
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…
In this paper, we give some {\L}ojasiewicz-type inequalities and a nonsmooth slope inequality on non-compact domains for continuous definable functions in an o-minimal structure. We also give a necessary and sufficicent condition for which…
In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of…
We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $\O_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation…
If F is an infinitely differentiable function whose composition with a blowing-up belongs to a Denjoy-Carleman class C_M (determined by a log convex sequence M=(M_k)), then F, in general, belongs to a larger shifted class C_N, where N_k =…
We prove a Nullstellensatz result for global ideals of Denjoy-Carleman functions in both finitely generated and infinitely generated cases.
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…
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…
We provide a new characterization of quasi-analyticity of Denjoy-Carleman classes, related to \emph{Wetzel's Problem}. We also completely resolve which Denjoy-Carleman classes carry \emph{sparse systems}: if the Continuum Hypothesis (CH)…
Inspired 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…