We give a new and simpler proof of a two-weight, weak $(p,p)$ inequality for fractional integrals first proved by Cruz-Uribe and Pérez [4].

A necessary and sufficient condition is given on the basis of a rare maximal function $M_l$ such that $M_{l}f\in L¹\left([0,1]\right)$ implies f ∈ L log L([0,1]).

We prove that for f ∈ L ln⁺L(ℝⁿ) with compact support, there is a g ∈ L ln⁺L(ℝⁿ) such that (a) g and f are equidistributed, (b) $M_S\left(g\right)\in L¹\left(E\right)$ for any measurable set E of finite measure.

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.

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.

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....

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.