Let $U₁,...,U_d$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $\mathbb{Z}{₊}^d$ and the associated collection of rectangular parallelepipeds in ${ℝ}^d$ with sides parallel to the axes and dimensions of the form $n₁\times \cdots \times n_d$ with $(n₁,...,n_d)\in \Gamma .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{\Gamma }$ are defined respectively on $L¹\left({ℝ}^d\right)$ and L¹(Ω) by $M_{}g\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|g\left(y\right)\right|dy$ and $M_{\Gamma }f\left(\omega \right)=su{p}_{(n₁,...,n_d)\in \Gamma }1/n₁\cdots n_d{\sum }_{j₁=0}^{n₁-1}\cdots {\sum }_{j_d=0}^{n_d-1}\left|f\left(U{₁}^{j₁}\cdots U_d^{j_d}\omega \right)\right|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate $|x\in {ℝ}^d:M_{}g\left(x\right)>\alpha |\le C_{}{\int }_{{ℝ}^d}\Phi (c_{}\left|g\right|/\alpha )$ for...

We show that the transference method of Coifman and Weiss can be extended to Hardy and Sobolev spaces. As an application we obtain the de Leeuw restriction theorems for multipliers.

By using the notion of contraction of Lie groups, we transfer $L^p-L²$ estimates for joint spectral projectors from the unit complex sphere $S^{2n+1}$ in ${ℂ}^{n+1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.

Nous étudions d’abord la transformation de Fourier sur les espaces ${\ell }^p(L^{p^{\text{'}}})$ qui sont formés de fonctions appartenant localement à $L^{p^{\text{'}}}$ et se comportant à l’infini comme des éléments de ${\ell }^p$. Si $1\le p,p^{\text{'}}\le 2$, les transformées de Fourier des éléments de ${\ell }^p(L^{p^{\text{'}}})$ appartiennent à ${\ell }^{q^{\text{'}}}(L^q)$. Dans les autres cas, nous donnons quelques résultats partiels.Nous montrons ensuite que ${\ell }^2(L^1)$ est le plus grand espace vectoriel solide de fonctions mesurables sur lequel la transformation de Fourier puisse se définir par prolongement par continuité.

We consider quadrature mirror filters, and the associated wavelet packet transform. Let X = {Xn}n∈Z be a stationary signal which has a continuous spectral density f. We prove that the 2n signals obtained from X by n iterations of the transform converge to white noises when n → +∞. If f is holderian, the convergence rate is exponential.

The process of translation averaging is known to improve dyadic BMO to the space BMO of functions of bounded mean oscillation, in the sense that the translation average of a family of dyadic BMO functions is necessarily a BMO function. The present work investigates the effect of translation averaging in other dyadic settings. We show that translation averages of dyadic doubling measures need not be doubling measures, translation averages of dyadic Muckenhoupt weights need not be Muckenhoupt weights,...

The single underlying method of averaging the wavelet functional over translates yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of known results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds by oversampling, and on equivalence of affine and quasiaffine frames. The method applies to multiwavelet systems in all dimensions, to dilation matrices that are in some cases not expanding, and to dual...

We study convolution operators bounded on the non-normable Lorentz spaces $L^{1,q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1,q}$. In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals,...

If $E\left(f\right)=\{x:lim\; supf☆{\mu }_j\left(x\right)>lim\; inff☆{\mu }_j\left(x\right)\}$, we examine the type of convergence of $g_k$ to $f$ so that $\left|E\right(g_k\left)\right|\le M$, $k=1,2,\cdots$, implies $\left|E\right(f\left)\right|\le M$.

The transplantation operators for the Hankel transform are considered. We prove that the transplantation operator maps an integrable function under certain conditions to an integrable function. As an application, we obtain the L¹-boundedness and H¹-boundedness of Cesàro operators for the Hankel transform.