On the expansion with respect to Steklov means of the second differences of functions

We shall prove that every function locally integrable in then-dimensional Euclidean spaceR ⁿ can be expanded into a series whose terms are the Steklov means of the second differences of the given function. In addition, the lengths of the edges of the cubes with respect to which averaging is taken form an infinite decreasing geometric progression. The series obtained in this way converge almost everywhere inR ⁿ . If the function expanded belongs to the Lebesgue spaceL _p on a compact set ofR ⁿ for some 1≤p<∞, then the expansion converges also in the norm of this space.     

The size of sums of sets,II

We prove inequalities which give lower bounds for the Lebesgue measures of setsE +K whereK is a certain kind of Cantor set. For example, ifC is the Cantor middle-thirds subset of the circle groupT, then $$m(E)^{1 - log2/log3} \leqq m(E + C)$$ for every BorelE ?T.     

A topological property enjoyed by near points but not by large points

Let H denote the halfline [0,∞). A point p?βH?H is called a near point if p is in the closure of some countable discrete closed subspace of H. In addition, a point p?βH?H is called a large point if p is not in the closure of a closed subset of H of finite Lebesgue measure. We will show that for every autohomeomorphism ? of βH?H and for each near point p we have that ?(p) is not large. In addition, we establish, under CH, the existence of a point x?βH?H such that for each autohomeomorphism ? of βH?H the point ?(x) is neither large nor near.     

Sur l'inégalité logarithmique de Sobolev

We prove a strenghtened form of Gross's logarithmic Sobolev inequality (see [1]) with a “remainder term” added to the left-hand side. We use this result to determine the cases of equality on Rⁿ equipped with Lebesgue measure.     

On the generalization of density topologies on the real line

The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ? _J generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.     

On m-th root metrics with special curvature properties

In this Note, we prove that every m-th root Finsler metric with isotropic Landsberg curvature reduces to a Landsberg metric. Then, we show that every m-th root metric with almost vanishing H-curvature has vanishing H-curvature.     

Generalized Lebesgue points for Sobolev functions

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space (X, d, μ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B(x, r) converge to f(x) when r converges to 0. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function f ∈ M ^s,p (X), 0 < s ≤ 1, 0 < p < 1, where X is a doubling metric measure space, has generalized Lebesgue points outside a set of \(\mathcal{H}^h\)-Hausdorff measure zero for a suitable gauge function h.     

Variational problems with several volume constraints on the level sets

We consider functionals of the kind on , and we study the problem , where consists of those functions u whose level sets satisfy certain volume constraints , where denotes Lebesgue measure, and , are given numbers. Examples show that this problem may have no solution, even for simple smooth F. As a consequence, we relax the constraint to , i.e. , and we show that the minimizers over exist and are H?lder continuous. Then we prove several existence theorems for the original problem, showing that, under suitable assumptions on the integrand function F, every minimizer over actually belongs to . Received: 31 January 2001 / Accepted: 23 February 2001 / Published online: 23 July 2001     

Some extremely amenable groupsQuelques groupes extrêmement moyennables

A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe's result, we show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group. To cite this article: T. Giordano, V. Pestov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 273–278.     

Lipschitzian Properties of Integral Functionals on Lebesgue Spaces $\bf{L_p, 1 \leqslant {p} < \infty}$

We give complete characterizations of integral functionals which are Lipschitzian on a Lebesgue space L _p with p ≠ ∞. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces. In every cases, the Lipchitzian properties of the integral functional can be described by growth conditions on the subdifferentials of the integrand which are equivalent to Lipschitzian properties of the integrand.     

Pointwise Convergence of the Calderón Reproducing Formula

In this paper, we study the pointwise convergence of the Calderón reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every $f\in L_{w}^{p}(\mathbb {R}^{d})$ with an $\mathcal{A}_{p}$ weight w, 1??p<??, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.     

Construction of pseudo-isometries for treelike hyperbolic 3-manifolds of infinite volume

We introduce a family of rigid hyperbolic 3-manifolds of infinite volume with possibly infinitely many ends: the treelike manifolds. These manifolds generalize a family of constructive non compact surfaces – the equational surfaces – for which the homeomorphism problem is decidable. The proof of rigidity relies firstly on Thurston's theorem of compactness of the Teichmüller space of acylindrical compact 3-manifolds, and secondly, on Sullivan's rigidity theorem. To cite this article: O. Ly, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On Parameter Depending Differential Systems Under Carathéodory Conditions

For every λ in a complex domain G, consider on some interval I the initial value problem y′(λ,x) = A(λ,x)y(λ,x) + b(λ,x), y(λ,x₀) - y₀. If this problem satisfies the Carathéodory conditions for every A, then there exist locally absolutely continuous and almost everywhere differentiable solutions y(λ,· ) of the initial value problem. In general, the union N of the exceptional sets N _λ ? I where y(λ, ·) is not differentiate or does not fulfill the differential equation, is not of Lebesgue measure zero. It will be shown that N is of Lebesgue measure zero provided that A and b are holomorphic with respect to λ and their integrals with respect to x are locally bounded on G × I.     

Domain decomposition for a finite volume method on non-matching grids

We are interested in a robust and accurate domain decomposition method with Robin interface conditions on non-matching grids using a finite volume discretization. We introduce transmission operators on the non-matching grids and define new interface conditions of Robin type. Under a compatibility assumption, we show the equivalence between Robin interface conditions and Dirichlet–Neumann interface conditions and the well-posedness of the global and local problems. Two error estimates are given in terms of the discrete H¹-norm: one in O(h^1/2) with operators based on piecewise constant functions and the other in O(h) (as in the conforming case) with operators using a linear rebuilding. Numerical results are given. To cite this article: L. Saas et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Measurable events indexed by products of trees

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T ₁,...,T _d ) of homogeneous trees and its level product ?T is the subset of the Cartesian product T ₁×...×T _d consisting of all finite sequences (t ₁,...,t _d ) of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product ?T of a vector homogeneous tree T. We show that, by refining the index set to the level product ?S of a vector strong subtree S of T, such families of events become highly correlated. An analogue of Lebesgue’s density Theorem is also established which can be considered as the “probabilistic” version of the density Halpern-Läuchli Theorem.     

Equivalence between mixed finite element and multi-point finite volume methods

We consider the lowest-order Raviart–Thomas mixed finite element method for elliptic problems on simplicial meshes in two or three space dimensions. This method produces saddle-point type problems for scalar and flux unknowns. We show how to easily eliminate the flux unknowns, which implies an equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. We describe the stencil of the final matrix and give sufficient conditions for its symmetry and positive definiteness. We present a numerical example illustrating the performance of the proposed method. To cite this article: M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Dimension and measure for semi-hyperbolic rational maps of degree 2

We prove that almost every non-hyperbolic rational map of degree 2 has at least one recurrent critical point. This estimate is optimal because the set of rational maps with all critical points non-recurrent is of full Hausdorff dimension. To cite this article: M. Aspenberg, J. Graczyk, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Some curvature properties of Cartan spaces with mth root metrics

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider a Cartan space with the mth root metric. We prove that every mth root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg, and weakly Berwald spaces, respectively. Then we show that the mth root Cartan space of almost vanishing H-curvature satisfies H?=?0.     

On the best Sobolev trace constant and extremals in domains with holes

We study the dependence on the subset A⊂Ω of the Sobolev trace constant for functions defined in a bounded domain Ω that vanish in the subset A. First we find that there exists an optimal subset that makes the trace constant smaller among all the subsets with prescribed and positive Lebesgue measure. In the case that Ω is a ball we prove that there exists an optimal hole that is spherically symmetric. In the case p=2 we prove that every optimal hole is spherically symmetric. Then, we study the behavior of the best constant when the hole is allowed to have zero Lebesgue measure. We show that this constant depends continuously on the subset and we discuss when it is equal to the Sobolev trace constant without the vanishing restriction.