María J. Carro

Recent developments in the theory of Lorentz spaces and weighted inequalities

Introduction Boundedness of operators on characteristic functions and the Hardy operator Lorentz spaces The Hardy-Littlewood maximal operator in weighted Lorentz spaces Bibliography Index.

Boundedness of some integral operators

We apply the expression for the norm of a function in the weighted Lorentz space, with respect to the disbibution function, to obtain as a simple consequence some weighted inequalities for integral operators

Weak-type weights and normable Lorentz spaces

We show that the Lorentz space A1(w) is a Banach space if and only if the Hardy-Littlewood maximal operator M satisfies a certain weak-type estimate. We also consider the case of general measures. Finally, we study some properties of several indices associated to these spaces.

Endpoint estimates from restricted rearrangement inequalities

Let T be a sublinear operator such that (Tf) ∗ (t) ≤ h(t, � f � 1) for some positive function h(t,s) and every function f such that � f � ∞ ≤ 1. Then, we show that T can be extended continuously from a logarithmic type space into a weighted weak Lorentz space. This type of result is connected with the theory of restricted weak type extrapolation and extends a recent result of Arias-de-Reyna concerning the pointwise convergence of Fourier series to a much more general context.

From Restricted Weak Type to Strong Type Estimates

Let be a sublinear operator such that for some positive function and every measurable set . Then it is shown that under some conditions on the operator , this restricted weak type estimate can be extended to the set of all functions such that , in the sense that . This inequality allows strong type estimates for to be obtained on several classes of spaces, such as logarithmic spaces and Lorentz spaces. A similar problem for operators acting on radial functions is also studied.

Transference results on weighted Lebesgue spaces

We present some transference results for a convolution operator with kernel K which is bounded from Lp0 (w0) into Lp1 (w1). Our results are a natural extension of theclassical results of Coifman an ...

Hardy spaces on Z

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of H p (Z N ) in terms of an atomic decomposition. Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms. In this paper, we give a positive answer to this question.

Real interpolation for divisible cones

We give necessary and sufficient conditions on a general cone of positive functions to satisfy the Decomposition Property (DP) introduced in [5] and connect the results with the theory of interpola ...

Interpolation of Limited and Weakly Compact Operators on Families of Banach Spaces

In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos–Peetre methods of interpolation for finite families.

Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces

Fil: Agora, Elona. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica; Argentina. Universidad de Barcelona; Espana