Papers by Alberto Saracco
We prove that elliptic tubes over properly convex domains D ⊂ RP n are C-convex and complete Koba... more We prove that elliptic tubes over properly convex domains D ⊂ RP n are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds.
Abstract. We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongl... more Abstract. We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in C n, generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly discrete sequences in strongly pseudoconvex domains, generalizing results obtained in the unit ball by Jevtić, Massaneda and Thomas, by Duren and Weir, and by MacCluer. In his celebrated solution of the corona problem in the disk, Carleson [C] introduced an important class of measures to study the structure of the Hardy spaces of the unit disk ∆ ⊂ C. Let A be a Banach space of holomorphic functions on a domain D ⊂ C n, and assume that A is contained in Lp (D) for some p> 0. A
matica dell’Universita ̀ degli studi di Parma. 01 Mag 08 – 28 Feb 09 Assegnista di ricerca per il... more matica dell’Universita ̀ degli studi di Parma. 01 Mag 08 – 28 Feb 09 Assegnista di ricerca per il progetto “Geometria e topologia algebrica, sim-plettica e complessa”, nel settore MAT/03 Geometria, presso l’Università di Milano Bicocca. Secondo classificato ad un concorso per 4 assegni di ricerca biennali in Matematica, nel dicembre 2007. 10 Mag 07 – 30 Apr 08 Assegnista di ricerca per il progetto “Analisi Complessa e Geometria”, nel settore MAT/03 Geometria, presso l’Universita ̀ di Roma II “Tor Vergata”.
Abstract. We study mapping properties of Toeplitz operators associated to a finite positive Borel... more Abstract. We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D ⊂ ⊂ Cn. In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space Ap(D) into Ar(D) with r> p, generalizing and making more precise results by Čučkovic ́ and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this setting results obtained by Kaptanoğlu for the unit ball.
We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex do... more We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.
2. Remarks on the proofs of theorems in [12] 3 3. An isomorphism theorem for semi 1-coronae 5
Abstract. Let A be a domain of the boundary of a strictly pseudoconvex domain Ω of Cn and M a smo... more Abstract. Let A be a domain of the boundary of a strictly pseudoconvex domain Ω of Cn and M a smooth, closed, maximally complex submanifold of A. We find a subdomain à of Ω, depending only on Ω and A, and a complex variety W ⊂ à such that bW ∩ A = M. Moreover, a generalization to analytic sets of depth at least 4 is given. 1.
We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes into... more We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes into account area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean and Gauss curvature, the classical counterexample by Schwarz-Peano to the definition of area is also analyzed.
Ergodic Theory and Dynamical Systems
We construct automorphisms of ${\mathbb C}^2$ , and more precisely transcendental Hénon maps, wit... more We construct automorphisms of ${\mathbb C}^2$ , and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank one. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):{\mathbb C}^2\rightarrow {\mathbb C}$ holomorphic.
Rendiconti del Circolo Matematico di Palermo Series 2, 2021
A Correction to this paper has been published 10.1007/s12215-020-00516-3.
This book is an introduction to the theory of holomorphic functions of several complex variables.... more This book is an introduction to the theory of holomorphic functions of several complex variables. It is based on the courses attended by students of mathematics at Scuola Normale Superiore of Pisa. Its treated subjects range from an advanced undergraduate course to a Ph.D. level. The book is largely divided into three parts. The first one, perhaps the most curricular, deals with the domains of holomorphy and their characterizations, through different notions of convexity (holomorphic convexity, Leviconvexity and pseudoconvexity) and the Cauchy-Riemann equation.The extension of this matter to complex spaces, known as the Oka-Cartan theory, is the content of the second part. This theory systematically makes use of the local analytic geometry and of the theory of sheaves and cohomology. The last part deals with the interplay between the theory of topological algebras and the theory of holomorphic functions. Some of the advanced results in the field are overviewed, sometimes without det...
We transfer several characterizations of hyperbolic convex domains given in a recent joint paper ... more We transfer several characterizations of hyperbolic convex domains given in a recent joint paper by Bracci and one of the authors to analogous one for C-convex domains.
The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case o... more The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of some new integral-geometric formulas concerning both the Euclidean and geodesic total curvature of spherical curves.
Advanced Topics in Mathematical Analysis
There are several interactions between the general theory of commutative topological algebras ove... more There are several interactions between the general theory of commutative topological algebras over C with unity and that of the holomorphic functions of one or several complex variables. This chapter is intended to be a review chapter of classical results in this subjects and a list of very difficult and long-standing open problems. A big reference section will provide the reader material to dwell into the proofs of theorems and understand more deeply the subjects. This chapter will be divided in two parts. In the first we will review known results on the general theory of Banach algebras, Frechet algebras, LB and LF algebras, as well as the open problems in the field. In the second part we will examine concrete algebras of holomorphic functions, again reviewing known results and open problems. We will mainly focus on the interplay between the two theories, showing how each one can be considered either an object of study per se or a mean to understand better the other subject.
ArXiv, 2020
Citations are getting more and more important in the career of a researcher. But how to use them ... more Citations are getting more and more important in the career of a researcher. But how to use them in the best possible way? This is a satirical paper, showing a bad trend currently happening in citation trends, due to intensive use of citation metrics. I am putting this on the arXiv and on Researchgate. Should you be interested to publish this paper on a journal of which you are editor, let me know.
Rendiconti del Circolo Matematico di Palermo (1952 -)
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Papers by Alberto Saracco