New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces.     

Riesz transforms through reverse Hölder and Poincaré inequalities

We establish an equidistribution theorem for the zeros of random holomorphic sections of high powers of a positive holomorphic line bundle. The equidistribution is associated with a family of singular moderate measures. We also give a convergence speed for the equidistribution.     

Normal approximation on Poisson spaces: Mehler’s formula,second order Poincaré inequalities and stabilization

We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincaré inequality, as well as on the use of Malliavin operators, of Stein’s method, and of an (integrated) Mehler’s formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order difference operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the k-nearest neighbour graph, (ii) intrinsic volumes of k-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be optimal, thus significantly improving previous findings in the literature. As a necessary step in our analysis, we also derive new lower bounds for variances of Poisson functionals.     

Modulus and the Poincaré inequality on metric measure spaces

The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces. Moreover, several competing definitions for the Poincaré inequality are shown to coincide, if the underlying measure is doubling. One such characterization considers only continuous functions and their continuous upper gradients, and extends work of Heinonen and Koskela. Applications include showing that the p-Poincaré inequality (with a doubling measure), for p1, persists through to the limit of a sequence of converging pointed metric measure spaces — this extends results of Cheeger. A further application is the construction of new doubling measures in Euclidean space which admit a 1-Poincaré inequality. Mathematics Subject Classification (2000):31C15, 46E35.     

L1-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces

Science China Mathematics - In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L1(ℝn). The singular integral estimates that it is possible to use for Lp,...     

Weak log-Sobolev and L p weak Poincaré inequalities for general symmetric forms

Weak log-Sobolev and L^p weak Poincaré inequalities for general symmetric forms are investigated by using newly defined Cheeger’s isoperimetric constants. Some concrete examples of ergodic birth-death processes are also presented to illustrate the results.     

Algebra properties for Sobolev spaces — applications to semilinear PDEs on manifolds

In this work, we aim to prove algebra properties for generalized Sobolev spaces W ^s,p ?? L ^?? on a Riemannian manifold (or more general homogeneous type space as graphs), where W ^s,p is of Bessel-type W ^s,p := (1+L)^?s/m (L ^p ) with an operator L generating a heat semigroup satisfying off-diagonal decays. We do not require any assumption on the gradient of the semigroup. Instead, we propose two different approaches (one by paraproducts associated to the heat semigroup and another one using functionals). We also study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non-bounded geometry) and even in euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.     

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.     

Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.     

A sharp form of Poincaré type inequalities on balls and spheres

Summary Letu be a real valued function on ann-dimensional Riemannian manifoldM ⁿ. We consider an inequality between theL ^q-norm ofu minus its mean value overM ⁿ and theL ^p-norm of the gradient ofu.The best constant in such inequality is exhibited in the following cases: i)M ⁿ is an open ball inIR ⁿ andp=1, 0<qn/(n–1); ii)M ⁿ is a sphere inIR ⁿ +1 and eitherp=1, 0<qn/(n–1) orp>n,q=.

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Sunto Siau una funzione a valori reali dafinita su una varietà riemannianan-dimensionaleM ⁿ. Si considera una disuguaglianza tra la normaL ^q diu meno il suo valor medio suM ⁿ e la normaL ^p del gradiente diu.Si determina la costante ottimale in tale disuguaglianza nei seguenti casi: i)M ⁿ è un disco aperto inIR ⁿ ep=1, 0<qn/(n–1); ii)M ⁿ è una sfera inIR ⁿ +1 ep=1, 0<qn/(n–1) oppurep>n,q=.

     

Hilbert function,generalized Poincaré series and topology of plane valuations

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

Hypersingular parameterized Marcinkiewicz integrals with variable kernels on Sobolev and Hardy-Sobolev spaces

Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.     

Uniformization and the Poincaré metric on the leaves of a foliation by curves

In this paper we prove that a holomorphic foliation by curves, on a complex compact manifoldM, whose singularities are non degenerated and whose tangent line bundle admits a metric of negative curvature, satisfies the following properties:(a): All leaves are hyperbolic.(b): The Poincaré metric on the leaves is continuous.(c): The set of uniformizations of the leaves by the Poincaré disc D is normal. Moreover, if ( _n ) _n 1 is a sequence of uniformizations which converges to a map : D, then either is a constant map (a singularity), or is an uniformization of some leaf. This result generalizes Theorem B of [LN], in which we prove the same facts for foliations of degree 2 on projective spaces.This research was partially supported by Pronex-Dynamical Systems, FINEP-CNPq.     

L~1-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L~1(R~n). The singular integral estimates that it is possible to use for L~p, p 1, are replaced here with inequalities which go back to Bourgain and Brezis(2007).