Nakhlé Asmar

On the distribution of Sidon series

Suppose that G is a compact abelian group with dual group F. Denote the normalized Haar measure on G by #. Let C(G) be the Banach space of continuous complex-valued functions on G. If S CF, a function f E L l ( G ) is called S-spectral whenever f is supported in S, where here and throughout the paper A denotes taking the Fourier transform. The collection of S-spectral functions that belong to a class of functions ]/Y will be denoted by l/Vs.

TRANSFERENCE OF STRONG TYPE MAXIMAL INEQUALITIES BY SEPARATION-PRESERVING REPRESENTATIONS

Improved dry cleaning formulation containing a dry cleaning solvent, water, inorganic polyphosphate salt, hydrogen peroxide and a suitable detergent surfactant having a pH value of from 5 to 9, which minimizes equipment corrosion and maintains fabric strength while effectively removing hydrophilic stains.

Representations of groups with ordered duals and generalized analyticity

Let G be a locally compact abelian group whose dual group G has a Haar-measurable order. We show that for each strongly continuous, uniformly bounded representation R of G in a UMD space X, there is a corresponding direct-sum decomposition of X which reflects the order in G. The projections in X corresponding to this direct-sum decomposition have norms controlled solely by the bound of R and a constant depending only on X. We illustrate how this “vector-valued harmonic conjugation” result generalizes the various abstract successors of the M. Riesz theorem and we introduce an application to the superdiagonalization of kernels for abstract integral operators.

Distributional control and generalized analyticity

Let S be a strongly continuous, separation-preserving representation of a locally compact abelian group G in Lp(μ), where 1≤p<∞, and μ is an arbitrary measure. We show that S is uniformly bounded with respect to the Lp-and L∞-norms if and only if it satisfies a certain boundedness condition for distribution functions. These equivalent conditions facilitate the transference from Lp(G) to Lp(μ) of the a.e. convergence for a wide class of sequences of convolution operators. The result unifies and generalizes various aspects of ergodic theory--in particular, the ergodic singular integral operators and ergodic Hardy spaces.

A note on UMD spaces and transference in vector-valued function spaces

A Banach space X is called an HT space if the Hilbert transform is bounded from Lp(X) into Lp(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in Lp(X), 1 < p < ∞. Berkson, Gillespie, and Muhly [5] showed that X ∈ HT =⇒ X ∈ ACF. In this note, we will show that X ∈ ACF =⇒ X ∈ UMD, thus providing a new proof of Bourgain’s result X ∈ HT =⇒ X ∈ UMD.

Maximal estimates on measure spaces for weak-type multipliers

We describe sufficient conditions for transferring from locally compact abelian groups to measure spaces the weak-type bounds of maximal operators defined by multipliers of weak type. This leads to homomorphism theorems for maximal multiplier operators.

On Jodeit’s multiplier extension theorems

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ℝ, and let ξ[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forLP (ℝ). The summability process corresponding to Δ extendsLP (T)-multipliers from ℤ to ℝ by linearity over the intervals [n, n + 1],n ∈ ℤ, when 1 ≤p < ∞, while the summability process corresponding to ξ[0,1) extends LP(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ℤ, when 1 <p < ∞. We describe how both these results have the following complete generalization: for 1 ≤p < ∞, an arbitrary compactly supported multiplier forLP (ℝ) will act as a summability kernel forLP (T)-multipliers, transferring maximal estimates from LP(T) to LP(ℝ). In particular, specialization of this maximal theorem to Jodeit’s summability kernel ξ[0, 1) provides a quick structural way to recover the fact that the maximal partial sum operator on LP(ℝ), 1 <p < ∞, inherits strong type (p,p)-boundedness from the Carleson-Hunt Theorem for Fourier series. Another result of Jodeit treats summability kernels lacking compact support, and we show that this aspect of multiplier theory sets up a lively interplay with entire functions of exponential type and sampling methods for band limited distributions.