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...

We apply a decomposition lemma of Uchiyama and results of the author to obtain good weighted Littlewood-Paley estimates for linear sums of functions satisfying reasonable decay, smoothness, and cancellation conditions. The heart of our application is a combinatorial trick treating m-fold dilates of dyadic cubes. We use our estimates to obtain new weighted inequalities for Bergman-type spaces defined on upper half-spaces in one and two parameters, extending earlier work of R. L. Wheeden and the author....

We prove that $|{\sum }_{k=1}^{n}sin\left((2k-1)x\right)/k|<Si\left(\pi \right)=1.8519...$ for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951).

Under certain conditions on a function space X, it is proved that for every $L^{\infty }$-function f with $\parallel f{\parallel }_{\infty }\le 1$ one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, $mes\phi \ne 1\le \varepsilon \parallel f{\parallel }_1$ and $\parallel \phi f{\parallel }_X\le const(1+log{\varepsilon }^{-1})$. For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of $L^{\infty }$-functions on ${ℝ}^n$ whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.

Se establece una estimación fina para el operador bilineal de Littlewood-Paley. Como aplicación se obtienen desigualdades para la norma ponderada y estimaciones del tipo L log L para el operador bilineal.

The best constant in the usual $L^p$ norm inequality for the centered Hardy-Littlewood maximal function on ${ℝ}^1$ is obtained for the class of all “peak-shaped” functions. A function on the line is called peak-shaped if it is positive and convex except at one point. The techniques we use include variational methods.

The aim of this paper is to obtain sharp estimates from below of the measure of the set of divergence of the m-fold Fourier series with respect to uniformly bounded orthonormal systems for the so-called G-convergence and λ-restricted convergence. We continue the study begun in a previous work.

We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍, $M_{\gamma }⨍$, by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of $M_{\gamma }$ between classical Lorentz spaces.