
Guillermo CurberaUniversity of Seville | US · Departamento de Análisis Matemático
Guillermo Curbera
Ph.D. Mathematics
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Publications (91)
The finite Hilbert transform T$T$, when acting in the classical Zygmund space LlogL$L\textnormal {log} L$ (over (−1,1)$(-1,1)$), was intensively studied in [8]. In this note, an integral representation of T$T$ is established via the L1(−1,1)$L^1(-1,1)$‐valued measure mL1:A↦T(χA)$m_{L^1}: A\mapsto T(\chi _A)$ for each Borel set A⊆(−1,1)$A\subseteq (...
The finite Hilbert transform $T$, when acting in the classical Zygmund space $\logl$ (over $(-1,1)$), was intensively studied in \cite{curbera-okada-ricker-log}. In this note an integral representation of $T$ is established via the $L^1(-1,1)$-valued measure $\mlog\colon A\mapsto T(\chi_A)$ for each Borel set $A\subseteq(-1,1)$. This integral repre...
Let X be a rearrangement-invariant space on [0, 1]. It is known that its Zippin indices $$\underline{\beta }{}_X,\overline{\beta }{}_X$$ β ̲ X , β ¯ X and its inclusion indices $$\gamma _X,\delta _X$$ γ X , δ X are related as follows: $$0\le \underline{\beta }{}_X\le 1/\gamma _X \le 1/\delta _X\le \overline{\beta }{}_X\le 1$$ 0 ≤ β ̲ X ≤ 1 / γ X ≤...
We show that inner functions are extreme points of the unit ball of the Hardy-Lorentz space H ( Λ ( φ ) ) H(\Lambda (\varphi )) , for Λ ( φ ) \Lambda (\varphi ) a Lorentz space with φ \varphi strictly increasing and strictly concave.
For a separable rearrangement invariant space X on [0, 1] of fundamental type we identify the set of all \(p\in [1,\infty ]\) such that \(\ell ^p\) is finitely represented in X in such a way that the unit basis vectors of \(\ell ^p\) (\(c_0\) if \(p=\infty \)) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a fo...
The finite Hilbert transform $T$ is a classical (singular) kernel operator which is continuous in every rearrangement invariant space $X$ over $(-1,1)$ having non-trivial Boyd indices. For $X=L^p$, $1<p<\infty$, this operator has been intensively investigated since the 1940's (also under the guise of the ``airfoil equation''). Recently, the extensi...
We consider the space $\mathcal{H}(ces_p)$ of all Dirichlet series whose coefficients belong to the Ces\`{a}ro sequence space $ces_p$, consisting of all complex sequences whose absolute Ces\`{a}ro means are in $\ell^p$, for $1<p<\infty$. It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the bounded...
We investigate convolution operators in the sequence spaces $d_p$, for $1\le p<\infty$. These spaces, for $p>1$, arise as dual spaces of the \ces 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 exh...
The generalized Ces\`{a}ro operators $\mathcal{C}_t$, for $t\in[0,1)$, introduced in the 1980's by Rhaly, are natural analogues of the classical Ces\`{a}ro averaging operator $\mathcal{C}_1$ and act in various Banach sequence spaces $X\subseteq {\mathbb C}^{{\mathbb N}_0}$. In this paper we concentrate on a certain class of Banach lattices for the...
The finite Hilbert transform T is a singular integral operator which maps the Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By extending the Parseval and Poincar\'e-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation $T(f)=g$ whenever the data function $...
We investigate convolution operators in the sequence spaces dp, for 1≤p<∞. These spaces, for p>1, arise as dual spaces of the Cesàro sequence spaces cesp thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve dp into dp. It turns out that such multiplier spaces exhibit features which ar...
For a separable rearrangement invariant space $X$ on $[0,1]$ of fundamental type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is finitely represented in $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a follow up o...
The generalized Cesàro operators Ct, for t∈[0,1), introduced in the 1980's by Rhaly, are natural analogues of the classical Cesàro averaging operator C1 and act in various Banach sequence spaces X⊆CN0. In this paper we concentrate on a certain class of Banach lattices for the coordinate-wise order, which includes all separable, rearrangement invari...
The finite Hilbert transform T:X→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T:X\rightarrow X$$\end{document} acts continuously on every rearrangement invariant sp...
William Henry Young was President of the International Mathematical Union (IMU) from 1929 to 1932. These were turbulent times for international cooperation in science in general, and mathematics in particular. The main cause of conflict was the International Research Council (IRC), which was created after the war by the Allied Powers with the aim o...
We investigate the spectrum and fine spectra of the finite Hilbert transform acting on rearrangement invariant spaces over (−1,1) with non-trivial Boyd indices, thereby extending Widom's results for Lp spaces. In the case when these indices coincide, a full description of the spectrum and fine spectra is given.
Let $E$ be a rearrangement invariant (r.i.) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty)$ such that $f^*\chi_{[0,1]}\in E$ and $f^*\chi_{[1,\infty)}\in L^2$. We reveal close connections between properties of the generalized Rosenthal's space, corresponding to the space $Z_E$, and the behaviour of i...
We investigate the spectrum and fine spectra of the finite Hilbert transform acting on rearrangement invariant spaces over $(-1,1)$ with non-trivial Boyd indices, thereby extending Widom's results for $L^p$ spaces. In the case when these indices coincide, a full description of the spectrum and fine spectra is given.
It is proved that the finite Hilbert transform $T\colon X\to X$, which acts continuously on every rearrangement invariant space $X$ on $(-1,1)$ having non-trivial Boyd indices, is already optimally defined. That is, $T\colon X\to X$ cannot be further extended, still taking its values in $X$, to any larger domain space.
The original version of the article contained mistakes in equations. The corrected equations are given below.
The finite Hilbert transform T is a classical (singular) kernel operator which is continuous in every rearrangement invariant space X over (–1, 1) having non-trivial Boyd indices. For X = Lp, 1 < p < ∞, this operator has been intensively investigated since the 1940’s (also under the guise of the “airfoil equation”). Recently, the extension and inve...
We consider the space H(ces p ) of all Dirichlet series whose coefficients belong to the Cesàro sequence space ces p , consisting of all complex sequences whose absolute Cesàro means are in ℓ p , for 1<p<∞. It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the boundedness of point evaluations. We i...
The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible such domain (i.e., its optimal domain). Some classical operators are already optimally defined (e.g., the Hilber...
Given a finite set F={f1,⋯,fk} of nonnegative integers (written in increasing order of magnitude) and a classical discrete family (pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant det(pfi(x+j−1))i,j=1,⋯,k. In this paper we prove an invariance property of this kind of Casorati determinants...
The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible such domain (i.e, its \textit{optimal domain}). Some classical operators are already optimally defined (e.g., th...
G.B. Guccia had an ambitious goal: developing the Circolo Matematico di Palermo into the international association of mathematicians. The pursuit of this aim was influenced by his two most important mathematical relationships: with Vito Volterra and with Henri Poincare. The time came to obtain a professorship, and to consolidate the Circolo Matemat...
The school and university education of G.B. Guccia are presented. He studied in the new educational system created when the Kingdom of Italy was founded. Sicily had a long story of cultivators of mathematics, which we briefly review. Shortly after G.B. Guccia entered university the most important encounter of his life occurred: he met the geometer...
We review three important processes: the foundation of national mathematical societies, in particular, the London Mathematical Society and the Société Mathématique de France; the creation of research mathematical journals, focusing on Acta Mathematica and its founder Gösta Mittag-Leffler; and the organization of mathematics at international level,...
From 1908 to 1914 is when the Circolo Matematico di Palermo and the Rendiconti experienced their maximum splendor. The society became the largest mathematical society in the world in 1914, and the journal acquired high international prestige by publishing first-rate results. The peak of this process was the celebration in 1914 of the 30th anniversa...
The background of Giovanni Battista Guccia is presented. First, a short account of the history of Sicily. Secondly, we discuss the impact of the discovery of the asteroid Ceres at the beginning of the nineteenth century by Giuseppe Piazzi from the Palermo Observatory. The family origins of G.B. Guccia are presented, as well as details on the family...
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...
This book examines the life and work of mathematician Giovanni Battista Guccia, founder of the Circolo Matematico di Palermo and its renowned journal, the Rendiconti del Circolo matematico di Palermo.
The authors describe how Guccia, an Italian geometer, was able to establish a mathematical society in Sicily in the late nineteenth century, which b...
We study conditions on Banach spaces close to (Formula presented.) guaranteeing the existence of Random Unconditional Convergence and Divergence systems. Special attention is given to the Haar system and to Cesàro spaces.
Given a finite set $F=\{f_1,\cdots ,f_k\}$ of nonnegative integers (written in increasing size) and a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant $\det(p_{f_i}(x+j-1))_{i,j=1,\cdots,k}$. In this paper we prove a nice invariant property for this kind of C...
We exhibit a large class of Banach function spaces which fail to have the Radon-Nikodym property.
We establish the weak Banach-Saks property for function spaces arising as the
optimal domain of an operator.
The Ces\`aro function spaces $Ces_p=[C,L^p]$, $1\le p\le\infty$, have
received renewed attention in recent years. Many properties of $[C,L^p]$ are
known. Less is known about $[C,X]$ when the Ces\`aro 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...
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 , for equivalent to , , is a system of Random Unconditional Convergence (RUC system).
We prove that the local version of Khintchine inequality holds in an rearrangement invariant function space {Mathematical expression} on [0,1] if and only if the lower dilation index of the fundamental function of {Mathematical expression} is positive. A further characterization is given, based on the uniform behavior in {Mathematical expression} o...
We introduce the spaces \({H^{p}_{uc}}\) consisting of all functions in the Hardy space \({H^p, 1 < p < \infty}\), whose Taylor series are unconditionally convergent and analyze the action of the Cesàro operator in these spaces. There is a related class of Banach sequence spaces \({N^p, 1 < p < \infty}\), arising from harmonic analysis, in which th...
We introduce and study the largest Banach lattice (for the coordinate-wise order) which is a solid subspace of \({\mathbb{C}^\mathbb{N}}\) and to which the classical Cesàro operator \({\mathcal{C}\colon\ell^p \to \ell^p}\) (a positive operator) can be continuously extended while still maintaining its values in ℓ
p
. Properties of this optimal Bana...
We construct a rearrangement invariant space X on [0,1] with the property that all bounded linear operators from ℓ p , 1<p<∞, to X are compact, but there exists a non-compact operator from ℓ ∞ 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...
We prove a weighted version of the well-known Khintchine inequality for rearrangement invariant norms.
We introduce and study the largest Banach space of analytic functions on the unit disc which is solid for the coefficient-wise order and to which the classical Cesar operator C: H-2 -> H-2 can be continuously extended, while still maintaining its values in H-2. Properties of this Banach space H(ces(2)) as well as a characterization of individual an...
We discuss a remarkable feature of the Cesàro averaging operator acting in certain sequence spaces.
We present a novel proof of the fact that the spectrum of the Cesàro operator acting in ℓ
p
, for 1 < p < ∞, consists of the closed disc centered at q/2 and with radius q/2, where q is the conjugate index of p.
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.
For each 1 ≤ p < ∞,the classical cesàro operator ℓ from the Hardy space Hp to itself has the property that there exist analytic functions f ∉ Hp with ℓ(f) ∈ Hp. This article deals with the identification and properties of the (Banach) space [ℓ, Hp] consisting of allanalytic functions that ℓ maps into Hp. It is shown that [ℓ, Hp] contains classical...
Given a probability space (Ω, μ) and a rearrangement invariant space X on [0,1], in certain situations inequalities for spaces of $${\mathbb {R}}$$-valued functions on Ω are equivalent to the boundedness of an associated operator T
K
: L
∞ ([0, 1]) → X generated by a kernel K ≥ 0 on the unit square (e.g. Sobolev type inequalities or Riesz potentia...
What is an ICM? For more than 110 years mathematicians have been gathering together in these particular reunions known as International Congress of Mathematicians. They are unique among any other scientific or intellectual meeting. What is the peculiar feature which explains their singularity? Oswald Veblen hinted an explanation at the opening of A...
Khintchin inequalities show that a.e. convergent Rademacher series belong to all spaces Lp
([0, 1]), for finite p. In 1975 Rodin and Semenov considered the extension of this result to the setting of rearrangement invariant spaces...
N
of functions having square exponential integrability plays a prominent role in this problem.
Another way of gaugi...
On Mean Ergodic Operators.- Fourier Series in Banach spaces and Maximal Regularity.- Spectral Measures on Compacts of Characters of a Semigroup.- On Vector Measures, Uniform Integrability and Orlicz Spaces.- The Bohr Radius of a Banach Space.- Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology.- Defining Lim...
Let X be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space Λ(R,X) of all measurable functions x such that x⋅h∈X for every a.e. converging series h=∑anrn∈X, where (rn) are the Rademacher functions. We study the situation when Λ(R,X) is a rearrangement invariant space different from L∞. Particular atten...
Let ν be a vector measure with values in a Banach space Z. The integration map $$I_\nu: L^1(\nu)\to Z$$, given by $$f\mapsto \int f\,d\nu$$ for f ∈ L
1(ν), always has a formal extension to its bidual operator $$I_\nu^{**}: L^1(\nu)^{**}\to Z^{**}$$. So, we may consider the “integral” of any element f
** of L
1(ν)** as I
ν
**(f
**). Our aim is to i...
Let X be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space Λ(R, X) of measurable functions x such that x · h ∈ X for every a.e. converging series h = a n r n ∈ X, where (r n) are the Rademacher functions. We characterize the situation when Λ(R, X) = L ∞ . We also discuss the behaviour of partial sums...
We will deal exclusively with the integration of scalar (i.e., ℝ or ℂ)-valued functions with respect to vector measures. The general theory can be found in [36, 37, 32], [44, Ch. I II] and [67, 124], for example. For applications beyond these texts we refer to [38, 66, 80, 102, 117] and the references therein, and the survey articles [33, 68]. Each...
We study extension of operators T: E→ L0([0, 1]), where E is an F–function space and L0([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic
measures and the corresp...
We review the history of the International Congress of Mathematicians (ICM). These congresses arose at the end of the 19-th century, being the last step on the professionalizing process of mathematical research. Its meetings have gathered all areas of mathematical research, exhibiting the best mathematics of the moment, and honouring the great math...
We review the history of the International Congress of Mathematicians (ICM). These congresses arose at the end of the 19-th century, being the last step on the professionalizing process of mathematical research. Its meetings have gathered all areas of mathematical research, exhibiting the best mathematics of the moment, and honouring the great math...
New features of the Banach function space Lwp(ν), that is, the space of all ν-scalarly pth power integrable functions (with 1⩽p∞ and ν any vector measure), are presented. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract p-convex Banach lattices.
Sobolev imbeddings (over suitable open subsets of Rn) can be extended from the classical Lp-setting to that of more gen- eral norms (required to be rearrangement invariant) on the underlying function spaces. This has been thoroughly studied in recent years and shown to be intimately connected to an associated kernel operator (of one variable). This...
Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedn...
We present an extrapolation theory that allows us to obtain, from weighted Lp inequalities on pairs of functions for p fixed and all A∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A∞ weights and also modular inequalities with A∞ weights. Vector-valued inequalities are obtained auto...
New features of the Banach function space L1w(v), that is, the space of all v-scalarly integrable functions (with v any vector measure), are exposed. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract Banach lattices. Applications are also given to the optimal domain of kernel operator...
Let X be a rearrangement invariant (r.i.) function space on [0,1]. We consider the Rademacher multiplicator space Λ(R,X) of measurable functions x such that xh∈X for every a.e. converging series h=∑anrn∈X, where (rn) are the Rademacher functions. We show that for a broad class of r.i. spaces X, the space Λ(R,X) is not r.i. In this case, we identify...
Let $E$ be a rearrangement invariant (r.i.) function space on ‘0, 1’. We consider the space $\Lambda({\cal R},E)$ of measurable functions $f$ such that $fg \in E$ for every a.e. converging series $g =\sum a_nr_n \in E$, where $(r_n)$ are the Rademacher functions. Curbera ‘4’ showed that, for a broad class of spaces $E$, the space $\Lambda({\cal R},...
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 bot...
The problem of finding optimal lattice domains for kernel operators with values in rearrangement invariant spaces on the interval [0,1] is considered. The techniques used are based on interpolation theory and integration with respect to C([0, 1])–valued measures.
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
nr
n∈ E, where (r
n) are the Rademacher functions. 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 unde...
We review the development of the theory of integra- tion with respect to a vector measure with values in a Banach space. The starting point is a 1955 paper by Bartle, Dunford and Schwartz where the authors consider the vector version of Riesz's Theorem on bounded linear functionals on spaces of continuous functions over a com- pact space. Next we a...
The Volterra convolution operator 6f(x) fl!x ! u(xfiy) f(y) dy, where u(() is a non-negative non- decreasing integrable kernel on (0, 1), is considered. Under certain conditions on the kernel u, the maximal Banach function space on (0, 1) on which the Volterra operator is a continuous linear operator with values in a given rearrangement invariant f...
Let X be a rearrangement invariant function space on [0,1] in which the Rademacher functions (rn) generate a subspace isomorphic to ℓ2. We consider the space Λ(R, X) of measurable functions f such that fgX for every function g=∑bnrn where (bn)ℓ2. We show that if X satisfies certain conditions on the fundamental function and on certain interpolation...
We consider the space L1(ν) of real functions which are integrable with respect to a measure ν with values in a Banach space X. We study type and cotype for L1(ν). We study conditions on the measure ν and the Banach space X that imply that L1(ν) is a Hilbert space, or has the Dunford-Pettis property. We also consider weak convergence in L1(ν).
We consider the space L(1)(nu) of real functions which are integrable with respect to a measure nu with values in a Banach space X. We study type and cotype for L(1)(nu). We study conditions on the measure nu and the Banach space X that imply that L(1)(nu) is a Hilbert space, or has the Dunford-Pettis property. We also consider weak convergence in...
We consider the space of real functions which are integrable with respect to a countably additive vector measure with values in a Banach space. In a previous paper we showed that this space can be any order continuous Banach lattice with weak order unit. We study a priori conditions on the vector measure in order to guarantee that the resulting L1...