Infimum-convolution description of concentration properties of product probability measures, with applications

This paper is devoted to the concentration properties of product probability measures μ=μ₁⊗?⊗μn, expressed in term of dimension-free functional inequalities of the form

     

Pseudometrics, distances and multivariate polynomial inequalities

We discuss three natural pseudodistances and pseudometrics on a bounded domain in based on polynomial inequalities.     

Integral inequalities for the Hilbert transform applied to a nonlocal transport equation

We prove several weighted inequalities involving the Hilbert transform of a function f(x) and its derivative. One of those inequalities,

is used to show finite time blow-up for a transport equation with nonlocal velocity.     

Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ^′-condition, and suppose that an additive variant of interpolation inequality holds

where , is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions and . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.     

The effect of domain topology on the number of positive solutions for singular elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities

In this paper we consider the critical singular equation involving the Caffarelli-Kohn-Nirenberg inequalities of the type

     

Global positivity estimates and Harnack inequalities for the fast diffusion equation

We investigate local and global properties of positive solutions to the fast diffusion equation u_t=Δu^m in the range (d−2)₊/d<m<1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; we use them to derive sharp global positivity estimates and a global Harnack principle. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak and elliptic Harnack inequalities. Our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow.     

Ingham-type inequalities for complex frequencies and applications to control theory

For complex valued sequences of the form ωn=an+ibn with an∈R and bn?0, we prove inequalities of the form , for all sequences {xn} with . We apply these to prove exact null-controllability for a class of hinged beam equations with mild internal damping with either boundary control or internal control.     

Second-order semilinear elliptic inequalities in exterior domains

We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients in exterior domains in . For W(x)?|x|^−σ at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.     

Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

We find necessary density conditions for Marcinkiewicz–Zygmund inequalities and interpolation for spaces of spherical harmonics in with respect to the L^p norm. Moreover, we prove that there are no complete interpolation families for p≠2.     

Finitely additive representation of spaces

Let be any atomless and countably additive probability measure on the product space with the usual σ-algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset such that can be isometrically isomorphically embedded as a closed subspace of L^p(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps simple and continuous functions on to their restrictions to T.     

Probabilistic n-normed spaces, -n-compact sets and -n-bounded sets

In this paper, first we define and study the probabilistic n-normed spaces and -n-compactness, also we prove some theorems and inequalities. In the next section we define -n-boundedness and prove some results in relation between -n-compact and -n-bounded sets in these spaces.     

On the best constants in noncommutative Khintchine-type inequalities

We obtain new proofs with improved constants of the Khintchine-type inequality with matrix coefficients in two cases. The first case is the Pisier and Lust-Piquard noncommutative Khintchine inequality for p=1, where we obtain the sharp lower bound of in the complex Gaussian case and for the sequence of functions . The second case is Junge's recent Khintchine-type inequality for subspaces of the operator space RC, which he used to construct a cb-embedding of the operator Hilbert space OH into the predual of a hyperfinite factor. Also in this case, we obtain a sharp lower bound of . As a consequence, it follows that any subspace of a quotient of (RC)^* is cb-isomorphic to a subspace of the predual of the hyperfinite factor of type III₁, with cb-isomorphism constant. In particular, the operator Hilbert space OH has this property.     

Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups

We study in this article the improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces and Besov spaces . When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in Rn and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p=1, these two techniques are not available anymore and following M. Ledoux (2003) [12], in Rn, we will treat here the critical case p=1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of improved Sobolev inequalities using weak-type Sobolev spaces.     

Sobolev–Hardy space with general weight

In this paper, it is defined the kth order Sobolev–Hardy space with norm

Then the corresponding Poincaré inequality in this space is obtained, and the results are given that this space is embedded in with weight and in with weight ^q/2 for 1q<2. Moreover, we prove that the constant of k-improved Hardy–Sobolev inequality with general weight is optimal. These inequalities turn to be some known versions of Hardy–Sobolev inequalities in the literature by some particular choice of weights.     

Probability measures and Milyutin maps between metric spaces

We prove that the functor of Radon probability measures transforms any open map between completely metrizable spaces into a soft map. This result is applied to establish some properties of Milyutin maps between completely metrizable spaces.     

Nonparametric denoising of signals with unknown local structure, I: Oracle inequalities

We consider the problem of pointwise estimation of multi-dimensional signals s, from noisy observations (y_τ) on the regular grid . Our focus is on the adaptive estimation in the case when the signal can be well recovered using a (hypothetical) linear filter, which can depend on the unknown signal itself. The basic setting of the problem we address here can be summarized as follows: suppose that the signal s is “well-filtered”, i.e. there exists an adapted time-invariant linear filter with the coefficients which vanish outside the “cube” {0,…,T}^d which recovers s₀ from observations with small mean-squared error. We suppose that we do not know the filter q^*, although, we do know that such a filter exists. We give partial answers to the following questions:

– is it possible to construct an adaptive estimator of the value s₀, which relies upon observations and recovers s₀ with basically the same estimation error as the unknown filter ?

– how rich is the family of well-filtered (in the above sense) signals?

We show that the answer to the first question is affirmative and provide a numerically efficient construction of a nonlinear adaptive filter. Further, we establish a simple calculus of “well-filtered” signals, and show that their family is quite large: it contains, for instance, sampled smooth signals, sampled modulated smooth signals and sampled harmonic functions.