Contents Foreword vii Introduction 1 1 Boundedness of operators on characteristic functions and the Hardy operator 3 1. died when he was only 39. He had just received his Ph.D. degree a few months earlier, under our supervision. He was really happy for all the new projects he had for the future, and so were we, since he was an extraordinary mathematician,… (More)
We study weak-type (1, 1) weighted inequalities for the fractional integral operator I α. We show that the fractional maximal operator M α controls these inequalities when the weight is radially decreasing. However, we exhibit some counterexamples which show that M α is not appropriate for this control on general weights. We do provide, nevertheless, some… (More)
We study the boundedness of bilinear convolutions operators with positive kernels. We prove both necessary and sufficient conditions and, by means of several counterexamples we show that near the endpoints the behavior of positive translation-invariant bilinear operators can be quite different than that of positive linear ones.
We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces.
We introduce a new decreasing rearrangement for functions defined on a homogeneous tree, which enjoys very intuitive and natural properties. The idea is to iterate, with respect to a particular ordering, the usual one dimensional rearrangement on each geodesic. After showing the canonicity of this definition and the axioms of symmetrization, we prove our… (More)
In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (, ). However, the question for multidimen-sional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the… (More)
The specificity and affinity of antibody-antigen interactions is a fundamental way to achieve reliable biosensing responses. Different proteins involved with dry eye dysfunction: ANXA1, ANXA11, CST4, PRDX5, PLAA and S100A6; were validated as biomarkers. In this work several antibodies were tested for ANXA1, ANXA11 and PRDX5 to select the best candidates for… (More)
PURPOSE The etiology of keratoconus (KC) and the factors governing its progression are not well understood. It has been suggested that this disease might be caused by biochemical alterations in the cornea; changes in the expression profiles of human aqueous humor (hAH) proteins have been observed in some diseases. To gain a new insight into the molecular… (More)
We prove the Lorentz-Shimogaki and Boyd theorems for the spaces Λ p u (w). As a consequence, we give the complete characterization of the strong boundedness of H on these spaces in terms of some geometric conditions on the weights u and w, whenever p > 1. For these values of p, we also give the complete solution of the weak-type boundedness of the… (More)