23 research outputs found
Abstract Ces\`aro spaces: Integral representations
The Ces\`aro function spaces , , have
received renewed attention in recent years. Many properties of are
known. Less is known about when the Ces\`aro operator takes its values
in a rearrangement invariant (r.i.) space other than . In this paper
we study the spaces via the methods of vector measures and vector
integration. These techniques allow us to identify the absolutely continuous
part of and the Fatou completion of ; to show that is
never reflexive and never r.i.; to identify when is weakly sequentially
complete, when it is isomorphic to an AL-space, and when it has the
Dunford-Pettis property. The same techniques are used to analyze the operator
; it is never compact but, it can be completely continuous.Comment: 21 page
A note on extreme points of the unit of Hardy-Lorentz spaces
We show that inner functions are extreme points of the unit ball of the
Hardy-Lorentz space , for a Lorentz
space with strictly increasing and strictly concave.Comment: This is the final version, to be published in the Proceedings of the
American Mathematical Societ
Fine spectra of the finite Hilbert transform in function spaces
We investigate the spectrum and fine spectra of the finite Hilbert transform
acting on rearrangement invariant spaces over with non-trivial Boyd
indices, thereby extending Widom's results for spaces. In the case when
these indices coincide, a full description of the spectrum and fine spectra is
given.Comment: 26 pages, 1 figure. Minor changes from previous version. This is the
final version, to be published in Advances in Mathematic
The finite Hilbert transform on
We present a detailed survey of recent developments in the study of the
finite Hilbert transform and its corresponding inversion problem in
rearrangement invariant spaces on
Giovanni Battista Guccia: pioneer of international cooperation in mathematics
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 by 1914 would grow to become the largest and most international in the world, with one of the most influential journals of the time. The book highlights the challenges faced by Guccia in creating an international society in isolated Palermo, and places Guccia’s activities in the wider European context through comparisons with the formation of the London Mathematical Society and the creation of Mittag-Leffler’s Acta Mathematica in Stockholm. Based on extensive searches in European archives, this scholarly work follows both historical and scientific treads, and will appeal to those interested in the history of mathematics and science in general