Papers by Guillermo P. Curbera
Journal of Mathematical Analysis and Applications
We investigate convolution operators in the sequence spaces d p , for 1 ≤ p < ∞. These spaces, fo... more We investigate convolution operators in the sequence spaces d p , for 1 ≤ p < ∞. These spaces, for p > 1, arise as dual spaces of the Cesàro sequence spaces ces p thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve d p into d p. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of ℓ p .
The Scientific Context
Giovanni Battista Guccia, 2018
We review three important processes: the foundation of national mathematical societies, in partic... more We review three important processes: the foundation of national mathematical societies, in particular, the London Mathematical Society and the Societe Mathematique de France; the creation of research mathematical journals, focusing on Acta Mathematica and its founder Gosta Mittag-Leffler; and the organization of mathematics at international level, discussing the origins of the International Congresses of Mathematicians.
Functiones et Approximatio Commentarii Mathematici, 2014
We construct a rearrangement invariant space X on [0, 1] with the property that all bounded linea... more We construct a rearrangement invariant space X on [0, 1] with the property that all bounded linear operators from p , 1 < p < ∞, to X are compact, but there exists a non-compact operator from ∞ to X. The techniques used allow to give a new proof of the characterization given by Hernández, Raynaud and Semenov of the rearrangement invariant spaces on [0, 1] for which the canonical embedding into L 1 ([0, 1]) is finitely strictly singular.
The Projects of Guccia: First Stage
The founding of the Circolo Matematico di Palermo and its journal, the Rendiconti del Circolo Mat... more The founding of the Circolo Matematico di Palermo and its journal, the Rendiconti del Circolo Matematico di Palermo are discussed. In order to understand the path that led to these two events, we follow G.B. Guccia’s post-doctoral journey in the summer of 1880 through Paris, Reims and London. Despite many initial difficulties, the early success of the society and the journal encouraged G.B. Guccia to lead the society towards internationalization.
William Henry Young, an Unconventional President of the International Mathematical Union
Fine spectra and compactness of generalized Cesàro operators in Banach lattices in CN0
Journal of Mathematical Analysis and Applications, 2021
Monatshefte für Mathematik, 2021
The finite Hilbert transform T : X Ñ X acts continuously on every rearrangement invariant space X... more The finite Hilbert transform T : X Ñ X acts continuously on every rearrangement invariant space X on p´1, 1q having non-trivial Boyd indices. It is proved that T cannot be further extended, whilst still taking its values in X, to any larger domain space. That is, T : X Ñ X is already optimally defined.
Annali di Matematica Pura ed Applicata (1923 -), 2019
Unfortunately, the original article contained few typesetting errors in the equations. The correc... more Unfortunately, the original article contained few typesetting errors in the equations. The corrected equations are given below. (1) Line 250 the equation should be: T (gT X (f n) + f n T X (g)) = (T X (f n))(T X (g)
Invariance properties of Wronskian type determinants of classical and classical discrete orthogonal polynomials
Journal of Mathematical Analysis and Applications, 2019
Abstract Given a finite set F = { f 1 , ⋯ , f k } of nonnegative integers (written in increasing ... more Abstract Given a finite set F = { f 1 , ⋯ , f k } of nonnegative integers (written in increasing order of magnitude) and a classical discrete family ( p n ) n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant det ( p f i ( x + j − 1 ) ) i , j = 1 , ⋯ , k . In this paper we prove an invariance property of this kind of Casorati determinants when the set F is substituted by the set I ( F ) = { 0 , 1 , 2 , ⋯ , max F } ∖ { max F − f : f ∈ F } . Our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators (Krall discrete polynomials). These polynomials are orthogonal with respect to certain Christoffel transforms of the classical discrete measures. By passing to the limit, this invariance property is extended to Wronskian type determinants whose entries are Hermite, Laguerre and Jacobi polynomials.
Journal of Mathematical Analysis and Applications, 2016
The Cesàro function spaces Ces p = [C, L p ], 1 ≤ p ≤ ∞, have received renewed attention in recen... more The Cesàro function spaces Ces p = [C, L p ], 1 ≤ p ≤ ∞, have received renewed attention in recent years. Many properties of [C, L p ] are known. Less is known about [C, X] when the Cesàro operator takes its values in a rearrangement invariant (r.i.) space X other than L p. In this paper we study the spaces [C, X] via the methods of vector measures and vector integration. These techniques allow us to identify the absolutely continuous part of [C, X] and the Fatou completion of
MATHEMATICA SCANDINAVICA, 2011
For each 1 ≤ p < ∞, the classical Cesàro operator C from the Hardy space H p to itself has the pr... more For each 1 ≤ p < ∞, the classical Cesàro operator C from the Hardy space H p to itself has the property that there exist analytic functions f / ∈ H p with C(f) ∈ H p. We discuss the (Banach) space C H p consisting of all analytic functions that C maps into H p. It is shown that C H p contains classical Banach spaces X of analytic functions, genuinely larger than the space H p , such that the operator C has a continuous H p-valued extension to X. An important feature of C H p is that it is the largest amongst all such spaces X.
On the existence of RUC systems in rearrangement invariant spaces
Mathematische Nachrichten, 2015
We characterize rearrangement invariant spaces X on [0, 1] with the property that each orthonorma... more We characterize rearrangement invariant spaces X on [0, 1] with the property that each orthonormal system in X which is uniformly bounded in some Marcinkiewicz space Mφα , for φα equivalent to log−1/α(e/t) , α>0 , is a system of Random Unconditional Convergence (RUC system).
Mathematical Proceedings of the Royal Irish Academy, 2007
The Weyl calculus for a pair A = (A 1 , A 2) of self-adjoint (n × n)-matrices, due to H. Weyl, as... more The Weyl calculus for a pair A = (A 1 , A 2) of self-adjoint (n × n)-matrices, due to H. Weyl, associates a matrix W A (f) to each smooth function f defined on R 2 in a linear but typically not multiplicative way. Letting c A (λ) := det((A 1 − λ 1 I) 2 + (A 2 − λ 2 I) 2) for λ ∈ R 2 denote the joint characteristic polynomial of the pair A, it is known, for n ≤ 3, that A 1 A 2 = A 2 A 1 if and only if W A (c A) = 0. It is an open question whether this is still true for n ≥ 4. Our aim here is to pursue two new approaches: the role of the canonical order structure for self-adjoint matrices; and topological invariants arising from continuity properties of the non-linear map (A, f) → W A (f).
Let E be a symmetric space on [0, 1]. Let Λ(R, E) be the space of measurable functions f such tha... more Let E be a symmetric space on [0, 1]. Let Λ(R, E) be the space of measurable functions f such that fg ∈ E for every almost everywhere convergent series g = b n r n ∈ E , where (r n) are the Rademacher functions. In [3] it was shown that, for a broad class of spaces E , the space Λ(R, E) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on E for Λ(R, E) to be order isomorphic to L ∞. This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which Λ(R, E) is order isomorphic to a symmetric space that differs from L ∞. The answer is positive for the Orlicz spaces E = L Φq with Φ q (t) = exp |t| q − 1 and 0 < q < 2.
Studia Mathematica, 2003
Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way.... more Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on [0, 1]. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its optimal domain. Several results are devoted to identifying the maximal rearrangement invariant space inside the optimal domain. The methods and techniques used involve interpolation theory, Banach function spaces and vector integration.
Banach Space Properties of L 1 of a Vector Measure
Proceedings of the American Mathematical Society, 1995
Positivity, 2010
In the setting of rearrangement invariant spaces, optimal Sobolev inequalities (via the gradient)... more In the setting of rearrangement invariant spaces, optimal Sobolev inequalities (via the gradient) are well understood. By means of an alternative functional, we obtain new Sobolev inequalities which are finer than (and not necessarily equivalent to) the ones mentioned above.
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Papers by Guillermo P. Curbera