Ana Bernardis

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.

Perturbations of the Haar wavelet by convolution

In this note we show that the standard convolution regularization of the Haar system generates Riesz bases of smooth functions for L2(R), providing in this way an alternative to the approach given by Govil and Zalik [Proc. Amer. Math. Soc. 125 (1997), 3363–3370]. The simplest compactly supported wavelet, the Haar function given by h := χ[0,1/2)−χ[1/2,1), generates by integer translations and dyadic dilations an orthonormal basis for the space L(R). In a recent paper Govil and Zalik [2] gave an ad hoc spline type regularization h, ε > 0, of the Haar wavelet h, in such a way that h produces by integer translations and dyadic dilations a Riesz basis for L(R) with bounds approaching 1 for ε → 0. A basis {fn, n ∈ Z} of L(R) is called a Riesz basis with bounds A and B if A ∑ n∈Z+ |cn| ≤ || ∑ n∈Z+ cnfn|| ≤ B ∑ n∈Z+ |cn| for every numerical sequence {cn, n ∈ Z} ∈ `. The aim of this note is to show that standard approximations of the identity provide good Riesz bases as regularizations of the Haar system. We shall denote by f the convolution of f with φε(x) = 1/εφ(x/ε), where φ is an appropriate integrable function. Concretely, we shall prove the following theorem. Theorem 1. Let m be a nonnegative integer and let φ be an even function with support in [−1, 1], φ ∈ W (R) = {φ ∈ L(R) : the m derivative φ of φ belongs to L(R)} and ∫ φ = 1. Let ε > 0 be such that Mε := 110 √ 3 (1 + ||φ||1) ε < 1. Then, h := h ∗ φε belongs to C(R), has support in [−ε, 1 + ε], and {hj,k(x) := 2h(2x − k) : j, k ∈ Z} is a Riesz basis of L(R) with bounds (1− √ Mε) and (1 + √ Mε). Notice that spline type regularizations can be obtained from adequate choices of φ, for example h is piecewise linear if we take φ = 12χ[−1,1]. Received by the editors January 11, 2000 and, in revised form, April 20, 2000. 2000 Mathematics Subject Classification. Primary 42C40.

Two weighted inequalities for convolution maximal operators

Let

Differential transforms of Cesàro averages in weighted spaces

\varphi\colon \mathbb{R} \to [0,\infty)

WEIGHTED WEAK-TYPE INEQUALITIES FOR GENERALIZED HARDY OPERATORS

an integrable function such that

Wavelet characterization of functions with conditions on the mean oscillation

\varphi\chi_{(-\infty,0)} = 0

The Cesàro maximal operator in dimension greater than one

and

On Haar bases for generalized dyadic Hardy spaces

\varphi

Comparison of Hardy–Littlewood and dyadic maximal functions on spaces of homogeneous type

is decreasing in

Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type

(0,\infty)