Nous définissons un produit renormalisé par ondelettes qui améliore, dans certains cadres fonctionnels, les propriétés du produit usuel de deux fonctions. Grâce à cette technique de renormalisation du produit nous obtenons une démonstration par ondelettes d'une version précisée du théorème du Jacobien. Finalement nous établissons le lien entre ce produit renormalisé par ondelettes et les paraproduits de J.M. Bony.

A simpler proof is given for the recent result of I. Labuda and the author that a series in the space L0 (lambda) is subseries convergent if each of its lacunary subseries converges.

Under mild conditions on the weight function K we characterize lacunary series in the so-called ${}_K$ spaces.

In this paper Lambert multipliers acting between $L^p$ spaces are characterized by using some properties of conditional expectation operator. Also, Fredholmness of corresponding bounded operators is investigated.

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure $R$ and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:$$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}\<+\infty \mathrm{forsome}0\<{\alpha }^{*}\<+\infty .$$In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend...

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure R and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition: $$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}<+\infty \mathrm{forsome}0<{\alpha }^{*}<+\infty .$$ In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result,...

We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction ${f|}_U$ is not in $L^q\left(U\right)$. When (X,μ) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p’s but nowhere q-integrable for some other q’s (0 < p,q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-González,...

We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the reduced first...

It is shown that the same probabilistic metric as used by Schweizer and Sklar to obtain all Lp space metrics can be used to derive the metrics of Orlicz spaces.