Recent Developments in Real and Harmonic Analysis: in Honor of Carlos Segovia.

CARLOS A. CABRELLI, ED.; JOSE LUIS TORREA, ED.

Birkhauser

Lugar: Boston, US; Año: 2009 p. 212

9780817645311

From December 12 to 15, 2005, a number of harmonic analysts from all over the world gathered in Buenos Aires, Argentina, for a conference organized to honor Carlos Segovia Ferna ́ndez on the occasion of his sixty-eighth birth- day for his mathematical contributions and services to the development of mathematics in Argentina.? The conference took place at the Instituto Argentino de Matema ́tica (IAM). The members of the advisory committee were Luis Caffarelli, Cristian Guti ́errez, Carlos Kenig, Roberto Mac ́ıas, Jos ́e Luis Torrea, and Richard Wheeden. The local organizing committee members were Gustavo Corach (Chair), Carlos Cabrelli, Eleonor Harboure, Alejandra Maestripieri, and Bea- triz Viviani. Unfortunately, Segovia was not able to attend due to health problems. The conference atmosphere was full of emotion, and many fond memories of Carlos were recalled by the participants. It was at this meeting that the idea crystallized of writing a mathematical tour of ideas arising around Segovia´s work. Unfortunately, one year after the conference, Carlos passed away and could not see this book finished. The book starts with a chronological description of his mathematical life, entitled ?Carlos Segovia Ferna ́ndez.? This comprehensive presentation of his original ideas, and even their evolution, may be a source of inspiration for many mathematicians working in a huge area in the fields of harmonic analy- sis, functional analysis, and partial differential equations (PDEs). Apart from this contribution, the reader will find in the book two different types of chap- ters: a group of surveys dealing with Carlos? favorite topics and a group of PDE works written by authors close to him and whose careers were influenced in some way by him. In the first group of chapters, we find the contribution by Hugo Aimar related to spaces of homogeneous type. Roberto Mac ́ıas and Carlos Segovia showed that it is always possible to find an equivalent quasi-distance on a given space of homogeneous type whose balls are spaces of homogeneous type. Aimar uses this construction to show a stronger version of the uniform reg- XIV Preface ularity of the balls. Two recurrent topics in the work of Carlos Segovia were commutators and vector-valued analysis, and this pair of topics is the sub- ject of the chapter by Oscar Blasco. He presents part of the work by Segovia related to commutators, and he extends it to a general class of Caldero ́n? Zygmund operators. The words ?Hardy, Lipschitz, and BMO? spaces were again recurrent in the work of Segovia. An analysis of the behaviour of the product of a function in some Hardy space with a function in the dual (Lip- schitz space) is made in the chapter by Aline Bonami and Justin Feuto. In the last fifteen years Segovia was very interested in applying some of his former ideas in Euclidean harmonic analysis to different Laplacians. He made some contributions to the subject, as can be observed in the publications list in- cluded in the present book. Along this line of thought is the chapter by Liliana Forzani, Eleonor Harboure, and Roberto Scotto. They review some aspects of this harmonic analysis related to the case of Hermite functions and poly- nomials. The last Ph.D. students of Segovia were introduced by him to the world of ?one-sided? operators, with special attention to weighted inequali- ties. Francisco Mart ́ın-Reyes, Pedro Ortega and Alberto de la Torre survey this subject in their chapter. As the authors say, they try to produce a more or less complete account of the main results and applications of the theory of weights for one-sided operators. In the second group of chapters, the reader will find the chapter by Luis Caffarelli and Aram Karakhanyan dealing with solutions to the porous media equation in one space dimension. Topics such as travelling fronts, separation of variables, and fundamental solutions are considered. The chapter by Sagun Chanillo and Juan Manfredi considers the problem of the global bound, in the space L2, of the Hessian of the solution of a certain second-order differential operator in a strictly pseudo-convex pseudo-Hermitian manifold. In the clas- sical case, this global bound can be seen as a ?Cordes perturbation method? of the boundedness of the iteration of the Riesz transforms. Well-posedness theory of the initial-value problem for the Kadomtsev?Petviashvili equations is treated in the chapter by Carlos Kenig; a connection with the Korteweg?de Vries equation is also discussed. A survey of recent results on the solutions and applications of the Monge?Amp`ere equation is written by Cristian Guti ́errez. We thank all the contributors of this volume for their willingness to col- laborate in this tribute to Carlos Segovia and his work. We are grateful to John Benedetto for inviting us to include our book in his prestigious series Applied and Numerical Harmonic Analysis, to Ursula Molter and Michael Shub for their proofreading and helpful comments, and to Tom Grasso and Regina Gorenshteyn from Birkha ̈user for their editorial help. Buenos Aires and Madrid Carlos Cabrelli June 2009 Jos ́e Luis Torrea