Un analogue d'un théorème de Hardy pour la transformation de DunklAn analogue of Hardy's theorem for the Dunkl transform

In this Note we give a generalization of Hardy's theorem for the Dunkl transform $\text{F}{}_{\mathrm{D}}$ on $\text{R}{}^{\mathrm{d}}$. More precisely, for all a>0, b>0 and p,q∈[1,+∞], we determine the measurable functions f such that and , where $\text{L}{}_{\mathrm{k}}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ are the L^p spaces associated with the Dunkl transform. To cite this article: L. Gallardo, K. Trimèche, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 849–854.     

The Neumann problem in the half-spaceLe problème de Neumannn dans le demi-espace

We study here the nonhomogeneous Neumann problem in the half-space $\text{R}{}^{\mathrm{N}}{}_{\mathrm{+}}$ with N?2. We give in L^p theory, with 1<p<∞, a basic existence and regularity results in weighted Sobolev spaces. To cite this article: C. Amrouche, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 151–156.     

Mesure unitarisante : algèbre de Heisenberg,algèbre de VirasoroUnitarizing measure: the Heisenberg algebra,the Virasoro algebra

Let $\text{R}{}^{\mathrm{2\infty }}$ be the infinite product of countably many copies of $\text{R}{}^2$. A Borelian probability measure on the infinite dimensional topological space $\text{R}{}^{\mathrm{2\infty }}$ which is unitarizing for the canonical representation of the infinite dimensional Heisenberg algebra is a Gaussian measure on $\text{R}{}^{\mathrm{2\infty }}$. To cite this article: H. Airault, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 787–792.     

The range of the derivative of a differentiable bumpL'image de la dérivée d'une fonction bosse différentiable

We study the range of the derivative of a Frechet differentiable bump. X is an infinite dimensional separable C^p-smooth Banach space. We first prove that any connected open subset of $\text{X}{}^1$ containing 0 is the range of the derivative of a C^p-bump. Next, analytic subsets of $\text{X}{}^1$ which satisfy a natural linkage condition are the range of the derivative of a C¹-bump. We find analogues of these results in finite dimensions. We finally show that $\text{f′(}\text{R}{}^2\text{)}$ is the closure of its interior, if f is a C²-bump on $\text{R}{}^2$. To cite this article: T. Gaspari, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 189–194.     

C1-algebras without idempotents

The simplest statement of the main results are these: Let π be a free group on 2 generators. Let Cπ be the complex ring and L¹π the ring extension to L¹ sums. Then L¹π contains no idempotents. Furthermore, if α ? Cπ, β?L¹π are nonzero, then αβ ≠ 0. The first result is in the direction of proving that a certain simple $\text{C}{}^1\text{-}\text{algebra}$ has no idempotents yielding a counter-example to the suggestion that simple $\text{C}{}^1\text{-}\text{algebras}$ are generated by their projections.     

Singular sets of Sobolev functionsEnsembles singuliers des fonctions de Sobolev

We are interested in finding Sobolev functions with “large” singular sets. Given $\text{N,k∈}\text{N}$, 1<p<∞, kp<N, for any compact subset A of $\text{R}{}^{\mathrm{N}}$, such that its upper box dimension is less than N?kp, we construct a Sobolev function $\text{u∈}\text{W}{}^{\mathrm{k,p}}\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N?kp. To cite this article: D. ?ubrini?, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.     

Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problemsEstimations de Carleman globales pour des solutions faibles de problèmes elliptiques avec condition de Dirichlet non homogène

We consider a general second order elliptic equation with right-hand side $\text{f+∑}{}_{\mathrm{j=0}}{}^{\mathrm{N}}\text{?f}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{j}}\text{∈}\text{H}{}^{\mathrm{?1}}\text{(Ω)}$ where $\text{f,f}{}_{\mathrm{j}}\text{∈}\text{L}{}^2\text{(Ω)}$ and Dirichlet boundary condition g∈H^1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L² norms of f and f_j and the H^1/2 norm of g. This estimate depends on two real parameters s and λ which are supposed to be large enough and is sharp with respect to the exponents of these parameters. This allows us to obtain, for example, sharper estimates on the pressure term in the linearized Navier–Stokes equations and it turns out to be very useful in the context of controllability problems. To cite this article: O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33–38.     

Rôle de l'espace de Besov B∞−1,∞ dans le contrôle de l'explosion éventuelle en temps fini des solutions régulières des équations de Navier–Stokes

Let $\text{u∈C([0,T}{}^1\text{[;L}{}^{\mathrm{n}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}{}^{\mathrm{n}}\text{)}$ be a maximal solution of the Navier–Stokes equations. We prove that u is C^∞ on $\text{]0,T}{}^1\text{[×}\text{R}{}^{\mathrm{n}}$ and there exists a constant $\text{ε}{}_1\text{>0}$, which depends only on n, such that if $\text{T}{}^1$ is finite then, for all $\text{ω∈S(}\text{R}{}^{\mathrm{n}}\text{)}{}^{\mathrm{n}}\text{,}$ we have To cite this article: R. May, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Frontière de Martin des marches aléatoires sur certains hypergroupes d-dimensionnelsMartin boundary of random walks on certain d-dimensional hypergroups

We give an integral representation formula for harmonic functions of Markov chains on $\text{N}{}^{\mathrm{d}}$ and $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$ which transition probability is invariant by translations of a hypergroup, product of polynomial hypergroups for $\text{N}{}^{\mathrm{d}}$ and product of Sturm–Liouville hypergroups for $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$. To cite this article: L. Godefroy, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1029–1034.     

Représentations de Gelfand–Graev pour les groupes réductifs non connexesGelfand–Graev representations for non connected reductive groups

Let G be a connected reductive group defined over $\text{F}{}_{\mathrm{q}}$ and let F be the corresponding Frobenius endomorphism. Let σ be a quasi-central rational automorphism of G. We define in this article Gelfand–Graev representations of the group $\text{G}{}^{\mathrm{F}}\text{=G}{}^{\mathrm{F}}\text{.〈σ〉}$ when σ is unipotent and when it is semi-simple. We show that they have similar properties to Gelfand–Graev representations of the group G^F. To cite this article: K Sorlin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 179–184.     

Sur les exemples de Lins Neto de feuilletages algébriquesOn Lins Neto's examples of algebraic foliations

We describe a family of quadratic homogeneous vector fields of $\text{C}{}^3$ having global meromorphic solutions that are not completely integrable. Under the standard projection, the orbits of these fields will give the degree two foliations of $\text{CP}{}^2$ recently constructed by Lins Neto in connection with the Poincaré problem. We will interpret these as quotients of linear flows on a product of elliptic curves. To cite this article: A. Guillot, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 747–750.     

Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–StokesNon-blowup at large times and stability for global solutions to the Navier–Stokes equations

Suppose there exists a global solution u to the incompressible Navier–Stokes equations, such that $\text{u∈}\text{C}{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{1/2}}\text{)}$. We prove that its $\text{H}\text{?}{}^{\mathrm{1/2}}$ norm goes to 0 at infinity. We next use this fact to control the $\text{L}{}^2{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{3/2}}\text{)}$ norm of u, and finally we prove that such a solution is stable. To cite this article: I. Gallagher et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 289–292.     

Non existence de solutions globales de certaines équations d'ondes non linéairesNonexistence of global solutions to some nonlinear wave equations

We study the semilinear wave equation u_tt?Δu=p^?k|u|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$, where p is a conformal factor approaching 0 at infinity. We prove that the solutions blow-up in finite time for small powers m, while having an arbitrarily long life-span for large m. Furthermore, we study the finite time blow-up of solutions for the class of quasilinear wave equations u_tt?Δu=p^?k|Lu|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$. To cite this article: M. Aassila, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 961–966.     

Approximation pour la distance de WassersteinApproximation for the Wasserstein distance

Let $\text{C}$ be a set of probability-measures and P a probability on $\text{R}{}^{\mathrm{d}}$. Under some conditions, we show that we have a solution to the approximation problem of P by $\text{C}\text{.}$ There exists a probability $\text{Q}{}_0\text{∈}\text{C}$, such that $\text{l(P,Q}{}_0\text{)?l(P,Q),}\text{?Q∈}\text{C}$ where l is the square of the Wasserstein distance. To cite this article: N. Belili, H. Heinich, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 537–540.     

Up to isometries,a deformation is a continuous function of its metric tensorAux isométries près,une déformation est une fonction continue de son tenseur métrique

If the Riemann–Christoffel tensor associated with a field of class $\text{C}{}^2$ of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset $\text{Ω?}\text{R}{}^3$, then this field is the metric tensor field associated with a deformation of class $\text{C}{}^3$ of the set $\text{Ω}$, uniquely determined up to isometries of $\text{R}{}^3$. We establish here that the mapping defined in this fashion is continuous, for ad hoc metrizable topologies. To cite this article: P.G. Ciarlet, F. Laurent, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 489–493.     

Estimées Ck pour les couronnes de convexes de type fini de Cn

C^k estimates for convex domains of finite type in $\text{C}{}^{\mathrm{n}}$ are known from Alexandre (C. R. Acad. Paris, Ser. I 335 (2002) 23–26). We now want to show the same result for annuli. Precisely, we show that for all convex domains D and D′ relatively compact of $\text{C}{}^{\mathrm{n}}$, of finite type m and m′ such that $\text{D}\text{?D′}$, for all q=1,…,n?2, there exists a linear operator $\text{T}{}^1{}_{\mathrm{q}}$ from $\text{C}{}_{\mathrm{0,q}}\text{(}\text{D′?D}\text{)}$ to $\text{C}{}_{\mathrm{0,q?1}}\text{(}\text{D′?D}\text{)}$ such that for all $\text{k∈}\text{N}$ and all (0,q)-form f, $\text{?}\text{?}$-closed of regularity C^k up to the boundary, $\text{T}{}^1{}_{\mathrm{q}}\text{f}$ is of regularity C^{k+1/max(m,m′)} up to the boundary and $\text{?}\text{?}\text{T}{}_{\mathrm{q}}{}^1\text{f=f}$. We fit the method of Diederich, Fisher and Fornaess to the annuli by switching z and ζ. However, the integration kernel will not have the same behavior on the frontier as in the Diederich–Fischer–Fornaess case and we have to alter the Diederich–Fornaess support function which will not be holomorphic anymore. Also, we take care of the so generated residual term in the homotopy formula and show that it is extremely regular so that solve the $\text{?}\text{?}$ problem for it will not be difficult. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A Dunford-Pettis theorem for L1H∞⊥

The spaces in the title are associated to a fixed representing measure m for a fixed character on a uniform algebra. It is proved that the set of representing measures for that character which are absolutely continuous with respect to m is weakly relatively compact if and only if each m-negligible closed set in the maximal ideal space of L^∞ is contained in an m-negligible peak set for H^∞. J. Chaumat's characterization of weakly relatively compact subsets in $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ therefore remains true, and $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ is complete, under the first conditions. In this paper we also give a direct proof. From this we obtain that $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ has the Dunford-Pettis property.     

Problème de Gleason et interpolation pour les fonctions hyper-analytiquesGleason's problem and interpolation for hyperholomorphic functions

We prove a Gleason type theorem in the setting of functions hyperholomorphic in the unit ball of $\text{R}{}^4$. We give an interpretation of the result in terms of pairs of functions defined in the unit ball of $\text{C}{}^2$. Finally we use the theorem to study the homogeneous interpolation problem in the setting of hyperholomorphic functions. To cite this article: D. Alpay, M. Shapiro, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 889–894.     

Un invariant d'isotopie lisse pour des surfaces dans une variété de dimension 4An invariant of smooth isotopy for surfaces in a 4-manifold

We construct an invariant of smooth isotopy for surfaces smoothly embeded in 4-manifolds. This invariant is used to distinguish smooth embeddings of tori or Klein bottles that are regular homotopic in $\text{C}{}^1\text{×}\text{C}{}^1$, and that have the same complement and the same fundamental rack. To cite this article: C. Darolles, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 811–815.