We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_{\alpha }$ maps weak weighted Orlicz$-\phi$ spaces into appropriate weighted versions of the spaces $BM{O}_{\psi }$, where $\psi \left(t\right)=t^{\alpha /n}{\phi }^{-1}(1/t)$. This generalizes known results about boundedness of $I_{\alpha }$ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha$ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_{\alpha }$ acting on weak$-L_{\phi }$ for $\phi$ of lower type equal or greater than $n/\alpha$, is the same as the one solving the problem for weak...

Dans le chapitre I on indique la croissance de $\omega$ et de la fonction convexe $C$ pour que de $log\left|F\right(z\left)\right|\le \pi \left|y\right|-\omega \left(\right|y\left|\right)$$(y=\mathrm{Im}z)...$

We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${\left|\phi ₙ\right|}^p$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

We give an overview of the behavior of the classical Hilbert Transform H seen as an operator on Lp(R) and on weak-Lp(R), then we consider other operators related to H. In particular, we discuss various versions of Discrete Hilbert Transform and Fourier multipliers periodized in frequency, giving some partial results and stating some conjectures about their sharp bounds Lp(R) → Lp(R), for 1 < p < ∞.

This Memoir studies Weil’s well-known Explicit Formula in the theory of prime numbers and its associated quadratic functional, which is positive semidefinite if and only if the Riemann Hypothesis is true. We prove that this quadratic functional attains its minimum in the unit ball of the $L^2$-space of functions with support in a given interval $\left[-t,t\right]$, and prove again Yoshida’s theorem that it is positive definite if $t$ is sufficiently small. The Fourier transform of the functional gives rise to a quadratic...

On décrit de diverses façons les fermetures respectives, dans l’espace $BMO\left({ℝ}^n\right)$ et dans sa version locale $bmo\left({ℝ}^n\right)$, de l’ensemble des fonctions à support compact et de l’ensemble des fonctions $C^{\infty }$ à support compact. Certains de ces résultats sont nouveaux; d’autres, considérés comme classiques, ne semblent pas avoir fait l’objet de publication. Des contre-exemples permettent de vérifier la diversité des sous-espaces considérés.

It is well known that the condition “f ∈ L¹ and f̂ ∈ L¹” is not sufficient to ensure the validity of the Poisson summation formula ∑f(k) = ∑f̂(k). We discuss here a stronger condition "$x^{a}f\in L^p$ and ${\xi }^{b}f̂\in L^q$" and see for which values of a and b the condition is sufficient.

On donne une démonstration nouvelle (et un peu plus générale) d’un théorème de J. Delsarte sur les fonctions moyenne-périodiques de deux variables.

In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$-almost periodic functions in ${ℝ}^n,$ slowly oscillating functions in ${ℝ}^n,$ and further analyze the recently introduced class of quasi-asymptotically $c$-almost periodic functions...