The logarithmic HLS inequality for systems on compact manifolds

We prove an optimal logarithmic Hardy-Littlewood-Sobolev inequality for systems on compact m-dimensional Riemannian manifolds, for any m?2. We show that a special case of the inequality, involving only two functions, implies the general case by using an argument from the theory of linear programing.     

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.).     

? 1-regularized linear regression: persistence and oracle inequalities

We study the predictive performance of ? ₁-regularized linear regression in a model-free setting, including the case where the number of covariates is substantially larger than the sample size. We introduce a new analysis method that avoids the boundedness problems that typically arise in model-free empirical minimization. Our technique provides an answer to a conjecture of Greenshtein and Ritov (Bernoulli 10(6):971–988, 2004) regarding the “persistence” rate for linear regression and allows us to prove an oracle inequality for the error of the regularized minimizer. It also demonstrates that empirical risk minimization gives optimal rates (up to log factors) of convex aggregation of a set of estimators of a regression function.     

An optimal logarithmic Sobolev inequality with Lipschitz constants

In this paper, we give an optimal logarithmic Sobolev inequality on Rn with Lipschitz constants. This inequality is a limit case of the Lp-logarithmic Sobolev inequality of Gentil (2003) [7] as p→∞. As a result of our inequality, we show that if a Lipschitz continuous function f on Rn fulfills some condition, then its Lipschitz constant can be expressed by using the entropy of f. We also show that a hypercontractivity of exponential type occurs in the heat equation on Rn. This is due to the Lipschitz regularizing effect of the heat equation.     

The parametrized family of metric Mahler measures

Let M(α) denote the (logarithmic) Mahler measure of the algebraic number α. Dubickas and Smyth, and later Fili and the author, examined metric versions of M. The author generalized these constructions in order to associate, to each point in t∈(0,∞], a metric version Mt of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions Mt, using them to present an equivalent form of Lehmer?s conjecture. We show that the function t?Mtt(α) is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph t?Mt(α) for rational α.     

Representation of certain logarithmic functions as polynomials with a small number of nonzero coefficients

We study an element of the ring of algebraic integers having the special form 1 ? zλ. We obtain a formula for calculating its logarithmic functions. Thus, we verify the conjecture that logarithmic functions of the element 1 ? zλ are polynomials in z such that z appears only in powers that are divisors of the number of the logarithmic function.     

Integrability of optimal mappings

In this paper, we study the integrability of optimal mappings T taking a probability measure μ to another measure g · μ. We assume that T minimizes the cost function c and μ satisfies some special inequalities related to c (the infimum-convolution inequality or the logarithmic c-Sobolev inequality). The results obtained are applied to the analysis of measures of the form exp(?|x|^α).     

Nonlinear diffusions, hypercontractivity and the optimal L-Euclidean logarithmic Sobolev inequality

The equation u_t=Δ_p(u^1/(p−1)) for p>1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal L^p-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the L^p-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton-Jacobi equation and also related to the L^p-Euclidean logarithmic Sobolev inequality is then stated.     

More on the duality conjecture for entropy numbers

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Grothendieck's theorem for noncommutative C1-algebras,with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C¹-algebra $\text{Ol}$. We prove that every “approximable” operator from $\text{Ol}$ into $\text{Ol}$¹ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C¹-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k_G.     

A (one-dimensional) free Brunn–Minkowski inequality

We present a one-dimensional version of the functional form of the geometric Brunn–Minkowski inequality in free (non-commutative) probability theory. The proof relies on matrix approximation as used recently by Biane and Hiai et al. to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered here from the Brunn–Minkowski inequality as in the classical case. The method is used to extend to the free setting the Otto–Villani theorem stating that the logarithmic Sobolev inequality implies the transportation cost inequality. To cite this article: M. Ledoux, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Suita conjecture and the Ohsawa-Takegoshi extension theorem

We prove a conjecture of N. Suita which says that for any bounded domain D in ? one has $c_{D}^{2}\leq\pi K_{D}$ , where c _D (z) is the logarithmic capacity of ??D with respect to z∈D and K _D the Bergman kernel on the diagonal. We also obtain optimal constant in the Ohsawa-Takegoshi extension theorem.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.     

ABC implies the radicalized Vojta height inequality for curves

The truncated or radicalized counting function of a meromorphic function counts the number of times that f takes a value a, but without multiplicity. By analogy, one also defines this function for numbers. In this sequel to [M. van Frankenhuijsen, The ABC conjecture implies Vojta's height inequality for curves, J. Number Theory 95 (2002) 289-302], we prove the radicalized version of Vojta's height inequality, using the ABC conjecture. We explain the connection with a conjecture of Serge Lang about the different error terms associated with Vojta's height inequality and with the radicalized Vojta height inequality.     

Some consequences of Schanuel's conjecture

We study the algebraic independence of two inductively defined sets. Under the hypothesis of Schanuel's conjecture we prove that the exponential power tower E and its related logarithmic tower L are linearly disjoint.     

Équations de Hamilton–Jacobi et inégalités entropiques généraliséesHamilton–Jacobi equations and inequalities for generalised entropies

We prove the equivalence between a general logarithmic Sobolev inequality and the hypercontractivity of a Hamilton–Jacobi equation. We also recover that this property imply a transportation inequality established by [5]. These results provide a natural generalization of the work performed in [3]. To cite this article: I. Gentil, F. Malrieu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 437–440.     

A correlation inequality for bipartite graphs

We conjecture an integral inequality for a product of functionsh(x _i ,y _j ) where the diagram of the product is a bipartite graphG. In particular, this inequality states that the random graph with fixed numbers of vertices and edges contains the asymptotically minimal number of copies ofG.     

Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L which is symmetric with respect to μ. We assume that L satisfies a generalized curvature dimension inequality as introduced by Baudoin and Garofalo (2009) [9]. Our goal is to discuss functional inequalities for μ like the Poincaré inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.     

Heat kernel analysis on semi-infinite Lie groups

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.