arXiv subjects
Publications and source records attributed to Anna Valette.
We give a proof of the o-minimal version of the Whitney Extension Theorem simplified as compared to the original ones. A new simplifying ingredient is a definable variant of Urysohn's lemma for class $\mathcal{C}^q$ (see Section 3).
Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure admits $\mathscr{C}^\infty$ cell decomposition, we show that the approximation can be required to be $\mathscr{C}^\infty$ and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems.
We prove that, for any closed semialgebraic subset $W$ of $\mathbb{R}^n$ and for any positive integer $p$, there exists a Nash function $f:\mathbb{R}^n\setminus W\longrightarrow (0, \infty)$ which is equivalent to the distance function from $W$ and at the same time it is $\Lambda_p$-regular in the sense that $|D^\alpha f(x)|\leq C d(x, W)^{1- |\alpha|}$, for each $x\in \mathbb{R}^n\setminus W$ and each $\alpha\in \mathbb{N}^n$ such that $1\leq |\alpha|\leq p$, where $C$ is a positive constant. In particular, $f$ is Lipschitz. Some applications of this result are given.
We establish that if a submanifold $M$ of $\mathbb{R}^n$ is definable in some o-minimal structure then any definable submanifold $N\subset \mathbb{R}^n$ which is $\mathscr{C}^\infty$ diffeomorphic to $M$, with a diffeomorphism $h:N\to M$ that is sufficiently close to the identity, must be $\mathscr{C}^\infty$ definably diffeomorphic to $M$. The definable diffeomorphism between $N$ and $M$ is then provided by a tubular neighborhood of $M$.
We first define the trace on a domain $\Omega$ which is definable in an o-minimal structure. We then show that every function $u\in W^{1,p}(\Omega)$ vanishing on the boundary in the trace sense satisfies Poincar\'e inequality. We finally show, given a definable family of domains $(\Omega_t)_{t\in \mathbb{R}^k}$, that the constant of this inequality remains bounded, if so does the volume of $\Omega_t$.
We prove that if $M\subset \mathbb{R}^n$ is a bounded subanalytic submanifold of $\mathbb{R}^n$ such that $B(x_0,\epsilon)\cap M$ is connected for every $x_0\in\overline{M}$ and $\epsilon>0$ small, then, for $p\in [1,\infty)$ sufficiently large, the space $C^\infty(\overline{M})$ is dense in the Sobolev space $W^{1,p}(M)$. We also show that for $p$ large, if $A\subset \overline{M}\setminus M$ is subanalytic then the restriction mapping $ C^\infty(\overline{M})\ni u\mapsto u_{|A}\in L^p(A)$ is continuous (if $A$ is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of $\overline{M}$ is dropped.
Let $\Omega$ be a subanalytic bounded open subset of $\mathbb{R}^n$, with possibly singular boundary. We show that given $p\in [1,\infty)$, there is a constant $C$ such that for any $u\in W^{1,p}(\Omega)$ we have $||u-u_{\Omega}||_{L^p} \le C||\nabla u||_{L^p},$ where we have set $u_{\Omega}:=\frac{1}{|\Omega|}\int_{\Omega} u.$
Efroymson's approximation theorem asserts that if $f$ is a $\mathcal{C}^0$ semialgebraic mapping on a $\mathcal{C}^\infty$ semialgebraic submanifold $M$ of $\mathbb{R}^n$ and if $\varepsilon:M\to \mathbb{R}$ is a positive continuous semialgebraic function then there is a $\mathcal{C}^\infty$ semialgebraic function $g:M\to \mathbb{R}$ such that $|f-g|<\varepsilon$. 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 $\mathcal{C}^\infty$ cell decomposition. We also establish approximation theorems for Lipschitz and $\mathcal{C}^1$ definable functions.
We work with semi-algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.
In the paper "Geometry of polynomial mapping at infinity via intersection homology" the second and third authors associated to a given polynomial mapping $F : \C^2 \to \C^2$ with nonvanishing jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of the mapping. We generalize this result.
For given polynomial map $F:\C^2\to\C^2$ with nonvanishing jacobian we associate a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.