We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_k$ such that the weighted ergodic averages corresponding to ${\mu }_{n_k}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions, the rate...

We solve the initial value problem for the diffusion induced by dyadic fractional derivative $s$ in ${ℝ}^{+}$. First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator....

We classify weights which map reverse Hölder weight spaces to other reverse Hölder weight spaces under pointwise multiplication. We also give some fairly general examples of weights satisfying weak reverse Hölder conditions.

We introduce generalized Campanato spaces ${}_{p,\varphi }$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then ${}_{p,\varphi }=BMO$. We give a characterization of the set of all pointwise multipliers on ${}_{p,\varphi }$.

Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for $\varphi :X\times {ℝ}_{+}\to {ℝ}_{+}$, we denote by $bm{o}_{\varphi ,p}\left(X\right)$ the set of all functions $f\in L_{loc}^p\left(X\right)$ such that $su{p}_{a\in X,r>0}1/\varphi (a,r)(1/\mu \left(B(a,r)\right){ʃ}_{B(a,r)}|f\left(x\right)-f_{B(a,r)}{{|}^{p}d\mu )}^{1/p}<\infty$, where B(a,r) is the ball centered at a and of...

We study polyhedral Dirichlet kernels on the n-dimensional torus and we write a fairly simple formula which extends the one-dimensional identity ${\sum }_{j=-N}^N{e}^{ijt}=sin\left((N+(1/2))t\right)/sin\left((1/2)t\right)$. We prove sharp results for the Lebesgue constants and for the pointwise boundedness of polyhedral Dirichlet kernels; we apply our results and methods to approximation theory, to more general summability methods and to Fourier series on compact Lie groups, where we write an asymptotic formula for the Dirichlet kernels.

Let w be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse Hölder inequality with exponents 0 < α < β. In the present paper, sharp estimates of the compositions of the power means ${}_{\alpha }w\left(x\right):={\left((1/x){\int }_0^x{w}^{\alpha }\left(t\right)dt\right)}^{1/\alpha }$, x > 0, are obtained for various exponents α. As a result, for the function w a property of self-improvement of summability exponents is established.

On décrit un problème naturel concernant la transformation de Fourier. Soient $f$, $\widehat{f}$ deux fonctions associées par celle-ci, positives pour $\mid x\mid \ge a$ et nulles en zéro. Quelle est la borne inférieure pour $a$ ? En dimension supérieure, même question, l’intervalle étant remplacé par la boule de rayon $a$. On montre l’existence d’une borne inférieure strictement positive, qui est estimée en fonction de la dimension. La dernière section montre que cette question est naturellement liée à la théorie des fonctions zêta....