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Let p 2 (0; 1), let v be a weight on (0; 1) and let p (v) be the classical Lorentz space, determined by the norm kfk p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. When p 2 (1; 1), this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if p (v) = ? p (v), where kfk ? p (v) := (R 1 0 (f… (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 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)
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.
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)
Associated to the class of restricted-weak type weights for the Hardy operator R p , we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms… (More)
We characterize the weighted Hardy's inequalities for monotone functions in R n +. In dimension n = 1, this recovers the classical theory of B p weights. For n > 1, the result was only known for the case p = 1. In fact, our main theorem is proved in the more general setting of partially ordered measure spaces.
Excessive subconjunctival scarring is the main reason of failure of glaucoma filtration surgery. We analyzed conjunctival and systemic gene expression patterns after non penetrating deep sclerectomy (NPDS). To find expression patterns related to surgical failure and their correlation with the clinical outcomes. This study consisted of two consecutive… (More)