Kato's inequality when Δu is a measure

We extend the classical version of Kato's inequality in order to allow functions u∈L¹_loc such that Δu is a Radon measure. This inequality has been recently applied by Brezis, Marcus, and Ponce to study the existence of solutions of the nonlinear equation ?Δu+g(u)=μ, where μ is a measure and $\text{g}\text{:}\text{R}\text{→}\text{R}$ is a nondecreasing continuous function. To cite this article: H. Brezis, A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Cheeger-harmonic functions in metric measure spaces revisited

Let (X,d,μ)$(X,d,\mu )$ be a complete metric measure space, with μ   a locally doubling measure, that supports a local weak L²$L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on (X,d,μ)$(X,d,\mu )$. Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.     

Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds

Let $L=\mathrm{\Delta }-\mathrm{\nabla }\varphi \cdot \mathrm{\nabla }$ be a symmetric diffusion operator with an invariant measure $\mu (\mathrm{d}x)={\mathrm{e}}^{-\varphi (x)}\mathrm{d}x$ on a complete non-compact Riemannian manifold M. We give the optimal conditions on “the m-dimensional Ricci curvature associated with L” so that various Liouville theorems hold for L-harmonic functions, and that the heat semigroup $P_t={\mathrm{e}}^{tL}$ has the $C_0$-diffusion property and is unique in $L^1(M,\mu )$. As applications, we give the optimal conditions for the uniqueness of the positive L-invariant measure and the $L^1$-uniqueness of the intrinsic Schrödinger operators on complete non-compact Riemannian manifolds. We also give a criterion for the finiteness of the total mass of the L-invariant measure and establish the Calabi–Yau volume growth theorem for the L-invariant measure on complete Riemannian manifolds on which “the m-dimensional Ricci curvature associated with L” is non-negative. This leads us to prove that if M is a complete Riemannian manifold with a finite L-invariant measure for which the associated m-dimensional Ricci curvature is non-negative, then M is compact. Moreover, we obtain an upper bound diameter estimate of such Riemannian manifolds by using the dimension of L, the total μ-volume of M and the upper bound of the μ-volume of geodesic balls of a fixed radius. Finally, using the variational formulae in Riemannian geometry, we give a new proof of the Bakry–Qian generalized Laplacian comparison theorem.     

An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift

We generalize the classical Rayleigh–Faber–Krahn inequality to the case of the Dirichlet Laplacian with a drift. We also solve some optimization problems for the principal eigenvalue of the operator $?\mathrm{\Delta }+v?\mathrm{?}$ in a fixed domain with a control of the drift v in $L^{\infty }$. To cite this article: F. Hamel et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Isoperimetric inequality via Lipschitz regularity of Cheeger-harmonic functions

Let (X,d,μ)$(X,d,\mu )$ be a complete, locally doubling metric measure space that supports a local weak L²$L^2$-Poincaré inequality. We show that optimal gradient estimates for Cheeger-harmonic functions imply local isoperimetric inequalities.     

Crossed-products extensions,of Lp-bounds for amenable actions

We will extend earlier transference results due to Neuwirth and Ricard from the context of noncommutative $L_p$-spaces associated with amenable groups to that of noncommutative $L_p$-spaces associated with crossed-products of amenable actions. Namely, if $m:G\to \mathbb{C}$ is a completely bounded Fourier multiplier on $L_p$, then it extends to the crossed-product with similar bounds provided that the action θ is amenable and trace-preserving. Furthermore, our construction also allows to extend G-equivariant completely bounded operators acting on the space part to the crossed-product provided that the generalized Følner sets of the action θ satisfy certain accretivity property. As a corollary we obtain stability results for maximal $L_p$-bounds over crossed products. We derive, using that stability results, an application to the boundedness of smooth multipliers in the $L_p$-spaces of group algebras.     

Analysis of random walks in dynamic random environments via L2-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on ${\mathbb{Z}}^d$. We assume that the environment has a stationary distribution $\mu$ and satisfies the Poincaré inequality w.r.t. $\mu$. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $\mu$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of $L^2$- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.     

Sharp Hardy–Sobolev inequalities

Let Ω be a smooth bounded domain in ${\mathbb{R}}^N$, $N?3$. We show that Hardy's inequality involving the distance to the boundary, with best constant ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$), may still be improved by adding a multiple of the critical Sobolev norm. To cite this article: S. Filippas et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Rationality of the vertex algebra VL+ when L is a non-degenerate even lattice of arbitrary rank

In this paper we prove that the vertex algebra $V_L^{+}$ is rational if L is a negative definite even lattice of finite rank, or if L is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices L, we show that the Zhu algebras of the vertex algebras $V_L^{+}$ are semisimple. This extends the result of Abe from [T. Abe, Rationality of the vertex operator algebra $V_L^{+}$ for a positive definite even lattice L, Math. Z. 249 (2) (2005) 455–484] which establishes the rationality of $V_L^{+}$ when L is a positive definite even lattice of finite rank.     

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datumExistence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

$\text{?△}{}_{\mathrm{p}}\text{u+b(x)|?u|}{}^{\mathrm{\lambda }}\text{=μ}\text{in}\text{Ω,}\text{u=0}\text{on}\text{?Ω,}$

where $\text{Ω}$ is a bounded open subset of $\text{R}{}^{\mathrm{N}}$ (N?2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on $\text{Ω}$ or a function in $\text{L}{}^1\text{(Ω)}$, λ?0 and b belongs to the Lorentz space $\text{L}{}^{\mathrm{N,1}}\text{(Ω)}$ or to the Lebesgue space $\text{L}{}^{\mathrm{\infty }}\text{(Ω)}$. We prove existence and uniqueness of renormalized solutions. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 757–762.     

Generalized differentiation and bi-Lipschitz nonembedding in L1

We consider Lipschitz mappings, $f:X\to V$, where X is a doubling metric measure space which satisfies a Poincaré inequality, and V is a Banach space. We show that earlier differentiability and bi-Lipschitz nonembedding results for maps, $f:X\to {\mathbb{R}}^N$, remain valid when ${\mathbb{R}}^N$ is replaced by any separable dual space. We exhibit spaces which bi-Lipschitz embed in $L^1$, but not in any separable dual V. For certain domains, including the Heisenberg group with its Carnot–Caratheodory metric, we establish a new notion of differentiability for maps into $L^1$. This implies that the Heisenberg group does not bi-Lipschitz embed in $L^1$, thereby proving a conjecture of J. Lee and A. Naor. When combined with their work, this has implications for theoretical computer science. To cite this article: J. Cheeger, B. Kleiner, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.