Hugo Aimar

Multiresolution approximations and unconditional bases on weighted Lebesgue spaces on spaces of homogeneous type

Starting from a slight modification of the dyadic sets introduced by M. Christ in [A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601-628] on a space of homogeneous type (X,d,@m), an MRA type structure and a Haar system H controlled by the quasi distance d, can be constructed in this general setting in such a way that H is an orthonormal basis for L^2(d@m). This paper is devoted to explore under which conditions on the measure @n, the system H is also an unconditional basis for the Lebesgue spaces L^p(d@n). As a consequence, we obtain a characterization of these spaces in terms of the H-coefficients.

On Riesz transforms and maximal functions in the context of Gaussian Harmonic Analysis

The purpose of this paper is twofold. We introduce a general maximal function on the Gaussian setting which dominates the Ornstein-Uhlenbeck maximal operator and prove its weak type (1,1) by using a covering lemma which is halfway between Besicovitch and Wiener. On the other hand, by taking as a starting point the generalized Cauchy-Riemann equations, we introduce a new class of Gaussian Riesz Transforms. We prove, using the maximal function defined in the first part of the paper, that unlike the ones already studied, these new Riesz Transforms are weak type (1,1) independently of their orders.

On dyadic nonlocal Schrödinger equations with Besov initial data

Abstract In this paper we consider the pointwise convergence to the initial data for the Schrodinger–Dirac equation i ∂ u ∂ t = D β u with u ( x , 0 ) = u 0 in a dyadic Besov space. Here D β denotes the fractional derivative of order β associated to the dyadic distance δ on R + . The main tools are a summability formula for the kernel of D β and pointwise estimates of the corresponding maximal operator in terms of the dyadic Hardy–Littlewood function and the Calderon sharp maximal operator.

Dyadic nonlocal diffusions in metric measure spaces

Abstract In this paper we solve the initial value problem for the nonlocal diffusion generated by the space fractional derivative induced by the dyadic tilings of M. Christ on a space of homogeneous type. We consider the problems of pointwise and norm convergence to the initial data. The main tool is the use of the Haar system induced by a dyadic tiling, which is actually the set of eigenfunctions for the fractional derivative operator.

Wavelet characterization of functions with conditions on the mean oscillation

The space BMO of those real functions defined on IR n for which the mean oscillation over cubes is bounded, appears in the pioneer works of J. Moser [13] and John-Nirenberg [9] in the early sixties as a tool for the study of regularity of weak solutions of elliptic and parabolic differential equations. Their main result, known today as John-Nirenberg Theorem, provides a characterization of BMO in terms of the exponential decay of the distribution function on each cube. Although the depth of this result, the space BMO only became well known in harmonic analysis after the celebrated Fefferman-Stein theorem of duality for the Hardy spaces: BMO is the dual of the Hardy space H 1. Since H 1 was already known to be the good substitute of L 1 for many questions in analysis, the space BMO was realized as the natural substitute of L ∞ in the scale of the Lebesgue spaces. Also due to Fefferman and Stein is the Littlewood-Paley type characterization of BMO in terms of the derivatives of the harmonic extension and Carleson measures (see for example [16]). This result has a discrete version: the Lemarie-Meyer characterization of BMO using wavelets, [10].

Heavy Tailed Approximate Identities and σ -stable Markov Kernels

The aim of this paper is to present some results relating the properties of stability, concentration and approximation to the identity of convolution through not necessarily mollification type families of heavy tailed Markov kernels. A particular case is provided by the kernels Kt obtained as the t mollification of Lσ(t) selected from the family ℒ={Lσ:Lσ̂(ξ)=e−|ξ|σ,0<σ<2}

The Dyadic Fractional Diffusion Kernel as a Central Limit

\mathcal {L}=\{L^{\sigma }: \widehat {L^{\sigma }}{(\xi )=e^{-|{\xi }|}}^{\sigma }, 0<\sigma <2\}

Affinity and Distance. On the Newtonian Structure of Some Data Kernels

, by a given function σ with values in the interval (0,2). We show that a basic Harnack type inequality, introduced by C. Calderón in the convolution case, becomes at once natural to the setting and useful to connect the concepts of stability, concentration and approximation of the identity. Some of the general results are extended to spaces of homogeneous type since most of the concepts involved in the theory are given in terms of metric and measure.

Nonlocal Schrödinger equations in metric measure spaces

We obtain the fundamental solution kernel of dyadic diffusions in ℝ+ as a central limit of dyadic mollification of iterations of stable Markov kernels. The main tool is provided by the substitution of classical Fourier analysis by Haar wavelet analysis.

On Haar bases for generalized dyadic Hardy spaces

Abstract Let X be a set. Let K(x, y) > 0 be a measure of the affinity between the data points x and y. We prove that K has the structure of a Newtonian potential K(x, y) = φ(d(x, y)) with φ decreasing and d a quasi-metric on X under two mild conditions on K. The first is that the affinity of each x to itself is infinite and that for x ≠ y the affinity is positive and finite. The second is a quantitative transitivity; if the affinity between x and y is larger than λ > 0 and the affinity of y and z is also larger than λ, then the affinity between x and z is larger than ν(λ). The function ν is concave, increasing, continuous from R+ onto R+ with ν(λ) < λ for every λ > 0