A Search for Best Constants in the Hardy-Littlewood Maximal Theorem.
A semi-discrete Littlewood-Paley inequality
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....
A set of orthogonal polynomials induced by a given orthogonal polynomial.
A sharp bound for a sine polynomial
We prove that 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).
A sharp correction theorem
Under certain conditions on a function space X, it is proved that for every -function f with one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, and . For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of -functions on whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.
A sharp estimate for bilinear Littlewood-Paley operator.
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.
A sharp estimate for the Hardy-Littlewood maximal function
The best constant in the usual norm inequality for the centered Hardy-Littlewood maximal function on 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.
A sharp estimate of the measure of the set of divergence of multiple orthogonal Fourier series
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.
A sharp rearrangement inequality for the fractional maximal operator
We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍, , by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of between classical Lorentz spaces.
A Sharper Stability Bound of Fourier Frames.
A simple observation about compactness and fast decay of Fourier coefficients.
A simple proof of a weighted inequality for the Hardy-Littlewood maximal operator in .
A simple proof of the atomic decomposition for , 0 < p ≤ 1
A Singular Convolution Equation In The Space Of Distributions
A singular convolution equation in the space of distributions.
A special Gaussian rule for trigonometric polynomials.
A stability result on Muckenhoupt's weights.
We prove that Muckenhoupt's A1-weights satisfy a reverse Hölder inequality with an explicit and asymptotically sharp estimate for the exponent. As a by-product we get a new characterization of A1-weights.
A stable method for the inversion of the Fourier transform in
A general method is given for recovering a function , , knowing only an approximation of its Fourier transform.
A strengthened form of a theorem of Wiener.
A strengthening of a theorem of Marcinkiewicz
We consider a problem of intervals raised by I. Ya. Novikov in [Israel Math. Conf. Proc. 5 (1992), 290], which refines the well-known theorem of J. Marcinkiewicz concerning structure of closed sets [A. Zygmund, Trigonometric Series, Vol. I, Ch. IV, Theorem 2.1]. A positive solution to the problem for some specific cases is obtained. As a result, we strengthen the theorem of Marcinkiewicz for generalized Cantor sets.