2000 Mathematics Subject Classification: 42B10, 43A32.In this paper we take the strip KL = [0, +∞[×[−Lπ, Lπ], where L is a positive integer. We consider, for a nonnegative real number α, two partial differential operators D and Dα on ]0, +∞[×] − Lπ, Lπ[. We associate a generalized Fourier transform Fα to the operators D and Dα. For this transform Fα, we establish an Lp − Lq − version of the Morgan's theorem under the assumption 1 ≤ p, q ≤ +∞.

For 1 ≤ q < ∞, let ${}_q\left(\right)$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes ${}_q\left(\right)$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...

The main observation of this note is that the Lebesgue measure μ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant ω ≥ μ, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter the original Turán-Nazarov inequality, they necessarily enter the definition of ω.

We prove the continuity of an oscillatory singular integral operator T with polynomial phase P(x,y) on an atomic space $H_P^1$ related to the phase P. Moreover, we show that the cancellation condition to be imposed on T holds under more general conditions. To that purpose, we obtain a van der Corput type lemma with integrability at infinity.

We prove a class of uncertainty principles of the form $\parallel S_{g}f{\parallel }_1\le C(\parallel x^{a}f{\parallel }_p+\parallel {\omega }^{b}f̂{\parallel }_q)$, where $S_{g}f$ is the short time Fourier transform of f. We obtain a characterization of the range of parameters a,b,p,q for which such an uncertainty principle holds. Counter-examples are constructed using Gabor expansions and unimodular polynomials. These uncertainty principles relate the decay of f and f̂ to their behaviour in phase space. Two applications are given: (a) If such an inequality holds, then the Poisson summation formula is valid...

We prove an x-ray estimate in general dimension which is a stronger version of Wolff's Kakeya estimate [12]. This generalizes the estimate in [13], which dealt with the n = 3 case.

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2...

En dimension 1 on analyse la fonction irrégulière $r\left(x\right)={\sum }_{n=1}^{\infty }{n}^{-p}sin\left(n^{p}x\right)$ (p entier ≥ 2) en un point $x_0$ de dérivabilité (π est un tel point) et on démontre que le terme d’erreur est un chirp de classe (1 + 1/(2p-2), 1/(p-1), (p-1)/p). La fonction r(x) est dans l’espace 2-microlocal $C_{x_0}^{s,s^{\text{'}}}$ si et seulement si s+s’ ≤ 1 - 1/p et ps+s’≤ p - 1/2. En dimension 2, on obtient en (π,π) l’existence d’un plan tangent pour la surface $z={\sum }_{m,n=1}^{\infty }{(m^2+n^2)}^{-\gamma }sin(m^{2}x+n^{2}y)$ dès que γ>1.

The notion of non-orthogonal multi-resolution analysis and its compatibility with differentiation (as expressed by the commutation formula) lead us to the construction of a multi-resolution analysis of L2(Rn)n which is well adapted to the approximation of divergence-free vector functions. Thus, we obtain unconditional bases of compactly supported divergence-free vector wavelets.

We study the problem of $L^p$-boundedness ($1\&lt;p\&lt;\infty$) of operators of the form $m(L_1,\cdots ,L_n)$ for a commuting system of self-adjoint left-invariant differential operators $L_1,\cdots ,L_n$ on a Lie group $G$ of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when $G$ is a homogeneous group and $L_1,\cdots ,L_n$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.