Given a doubling measure μ on Rd, it is a classical result of harmonic analysis that Calderón-Zygmund operators which are bounded in L2(μ) are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on μ by a mild growth condition on μ. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderón-Zygmund decomposition adapted...

Let $\mu$ be a Fourier-Stieltjes transform, defined on the discrete real line and such that the corresponding measure on the dual group vanishes on the set of characters, continuous on $\mathbf{R}$. Then for every $ϵ\>0$, $\{x\in \mathbf{R}|\mathrm{Re}\left(\mu \right(x\left)\right)\>ϵ\}$ has a vanishing interior Lebesgue measure. If $ϵ=0$ the statement is not generally true. The result is applied to prove a theorem of Rosenthal.

Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator $S_d$ and its associated global maximal operator $S*{*}_d$ by $\left(S_{d}f\right)(x,t)=1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S*{*}_{d}f)\left(x\right)=su{p}_{t\in ℝ}|1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi |$, f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, $S_{d}f$ is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥ 2n/(n-2), the...

It is proved that, for some reverse doubling weight functions, the related operator which appears in the Fefferman Stein's inequality can be taken smaller than those operators for which such an inequality is known to be true.

A condition on a scaling function which generates a multiresolution analysis of $L^p\left({ℝ}^d\right)$ is given.

We study the behaviour of the $n$-dimensional centered Hardy-Littlewood maximal operator associated to the family of cubes with sides parallel to the axes, improving the previously known lower bounds for the best constants $c_n$ that appear in the weak type $(1,1)$ inequalities.

We prove the div-curl lemma for a general class of function spaces, stable under the action of Calderón-Zygmund operators. The proof is based on a variant of the renormalization of the product introduced by S. Dobyinsky, and on the use of divergence-free wavelet bases.

A proof of a necessary and sufficient condition for a sequence to be a multiplier of the normalized Haar basis of L¹[0,1] is given. This proof depends only on the most elementary properties of this system and is an alternative proof to that recently found by Semenov & Uksusov (2012). Additionally, representations are given, which use stochastic processes, of this multiplier norm and of related multiplier norms.

We prove a restriction theorem for the class-1 representations of the Heisenberg motion group. This is done using an improvement of the restriction theorem for the special Hermite projection operators proved in [13]. We also prove a restriction theorem for the Heisenberg group.

The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let $f\in C^{\alpha }(ℝ,ℝ)$ be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator $\left(A_{x}f\right)\left(r\right)=1/2r{\int }_{x-r}^{x+r}f\left(z\right)dz$ has a critical point at r = γ for every x ∈ ℝ, then f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ. This statement can be used to derive a characterization of trigonometric functions as those nonconstant...