Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg

En utilisant l'inégalité de Poincaré et la formule de représentation, on montre que sur le groupe de Heisenberg de dimension réelle 3, H¹, il existe une constante C>0 telle que :

     

Terme constant de fonctions sur un espace symétrique réductif p-adique

We generalize Casselman's pairing to p-adic reductive symmetric spaces and study the asymptotic behaviour of certain generalized coefficients. We also prove an analogue of a lemma due to Langlands which allows us to prove a disjunction result for the Cartan decomposition of the p-adic reductive symmetric spaces.     

Estimation des projecteurs de De Rham Hodge de certaines variétés riemaniennes non compactes

On montre des estimées pour le projecteur de De Rham–Hodge. We prove estimates for the De Rham–Hodge L^p projectors. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Étude de l'algèbre de Lie double des arbres enracinés décorés

The double Lie algebra LD of rooted trees decorated by a set D is introduced, generalising the construction of Connes and Kreimer. It is shown that it is a simple Lie algebra. Its derivations and its automorphisms are described, as well as some central extensions. Finally, the category of lowest weight modules is introduced and studied.     

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p−L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature hypothesis.     

Existence globale pour l'équation de Smoluchowski continue non homogène et comportement asymptotique des solutions

We prove global existence of solutions to the continuous nonhomogeneous Smoluchowski equation for coagulation rates satisfying a more general structure condition than the Galkin–Tupchiev monotony hypothesis considered in (Ph. Laurençot, S. Mischler, Arch. Rational Mech. Anal. 162 (1) (2002) 45–99). The Smoluchowski coagulation rate fulfils this condition as well as some rates which vanish on the diagonal. Under the condition of positivity of the coagulation rate outside of the diagonal we prove that solutions tend to 0 in the large time asymptotic. These results depend on a new estimate from below for the dissipation rate of the L^p-norm, p>1. To cite this article: S. Mischler, M. Rodriguez Ricard, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

Classes of singular integral operators along variable lines

We prove estimates for classes of singular integral operators along variable lines in the plane, for which the usual assumption of nondegenerate rotational curvature may not be satisfied. The main L^p estimates are proved by interpolating L² bounds with suitable bounds in Hardy spaces on product domains. The L² bounds are derived by almost-orthogonality arguments. In an appendix we derive an estimate for the Hilbert transform along the radial vector field and prove an interpolation lemma related to restricted weak type inequalities.     

Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms

For all d?2 and p∈(1,max(2,(d+1)/2)], we prove sharp Lp to Lp(Lq) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp((logt)P), where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d?2 and p∈(1,2]. As corollaries, we prove Lp bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in Rd.     

On the Aleksandrov–Bakelman–Pucci estimate for the infinity Laplacian

We prove L ^∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and p-Laplacian, namely $$\begin{array}{ll} -\Delta_p^N u=f\quad{\rm for } \; n < p \leq\infty.\end{array}$$ We are able to provide a stable family of results depending continuously on the parameter p. We also prove the failure of the classical Alexandrov–Bakelman–Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.     

Gaussian bounds for propagators perturbed by potentials

We develop the perturbation theory for propagators, with the objective to prove Gaussian bounds. Let U be a strongly continuous propagator, i.e., a family of operators describing the solutions of a non-autonomous evolution equation, on an Lp-space, and assume that U is positive and satisfies Gaussian upper and lower bounds. Let V be a (time-dependent) potential satisfying certain Miyadera conditions with respect to U. We show that then the perturbed propagator enjoys Gaussian upper and lower bounds as well. To prepare the necessary tools, we extend the perturbation theory of strongly continuous propagators and the theory of absorption propagators.     

La somme des faux degrés—un mystère en théorie des invariants

We continue the study of the polynomiality of the semicentre Sz(p) of the enveloping algebra of a parabolic subalgebra p of a semisimple Lie algebra g, motivated by its truth when g is of type A or C [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155-191] and when p=b, a Borel subalgebra [A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977) 241-289] and p=g (Chevalley).We construct a linear map of Sz(b) into Sz(g) and show it to be an isomorphism just in types A and C. We link this to the difficulty of proving the polynomiality of Sz(p) outside types A and C. It leads to “false degrees” defined by underlying combinatorial structure. These are the true degrees when the bounds in [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155-191] coincide and polynomiality ensues. We show that these false degrees always sum to which can fail for the true degrees when they are defined. Finally we prove the Tauvel-Yu conjecture on the index of a parabolic.     

Descente étale des F-isocristaux surconvergents et rationalité des fonctions L de schémas abéliens

Let k be a field of characteristic p>0 and S a smooth separated k-scheme; three types of results are proved:(1) etale descent for overconvergent F-isocrystals on S;(2) full faithfulness of restriction functors to a dense open subscheme of S, between F-isocrystals;(3) a p-adic cohomological expression of the L-function of an abelian scheme on S and its rationality.     

Approximation in p-Norm of Univariate Concave Functions

We derive worst-case bounds, with respect to the L ^p norm, on the error achieved by algorithms aimed at approximating a concave function of a single variable, through the evaluation of the function and its subgradient at a fixed number of points to be determined. We prove that, for p larger than 1, adaptive algorithms outperform passive ones. Next, for the uniform norm, we propose an improvement of the Sandwich algorithm, based on a dynamic programming formulation of the problem.     

Interrelation between the convergence rates in von Neumann’s and Birkhoff’s ergodic theorems

In the L _p spaces, 1 < _p < ∞, we prove some inequalities for discrete and continuous times that make it possible to obtain the convergence rate in Birkhoff’s theorem in the presence of bounds on the convergence rate in von Neumann’s ergodic theorem belonging to a sufficiently large rate range. The exact operator analogs of these inequalities for contraction semigroups in L _p are given. These results also have the obvious exact analogs in the class of wide-sense stationary stochastic processes.     

Approximation de compacts fonctionnels par des variétés analytiques

In the theory of approximation there are some problems on approximation of compact sets in functional spaces by analytic families. First, we deal with the case of algebraic varieties, the theorem of Vitushkin, in which we give a new proof based on the method of Warren, with precision of constants. Next, we consider the case of analytic varieties which is as well a negative result: we show that an analytic family with N variables cannot approach the compact Λl,s better than order as N increases. We finish by giving some applications in Sturm-Liouville inverse theory.     

Uniform bounds from bounded Lp-functionals in reaction-diffusion equations

To prove global existence of classical or mild solutions of reaction-diffusion equations, a priori bounds in the uniform norm are needed. But for interesting examples, often one can only derive bounds for some L_p-norms. Using the structure of the reaction term they can be used to obtain uniform bounds. Two propositions are stated which give conditions for this procedure. The proofs use the smoothing properties of analytic semigroups and the multiplicative Gagliardo-Nirenberg inequality. To illustrate the method, we prove global existence of solutions for the Brusselator and a Volterra-Lotka system with one diffusing and one sedentary species.     

An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations

We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L ²- and H ¹-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition.     

The general optimal L-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations

We prove a general optimal L^p-Euclidean logarithmic Sobolev inequality by using Prékopa-Leindler inequality and a special Hamilton-Jacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (J. Funt. Anal.).     

An <Emphasis Type="Italic">h</Emphasis>-<Emphasis Type="Italic">p</Emphasis> version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations

We consider an h-p version of the continuous Petrov-Galerkin time stepping method for Volterra integro-differential equations with proportional delays. We derive a priori error bounds in the L ²-, H ¹- and L ^∞-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Numerical experiments are presented to illustrate the theoretical results.