Advanced Courses of Mathematical Analysis V

The following sections are included:

• PREFACE
• SPONSORS
• ORGANIZING COMMITTEES
• CONTENTS

The following pages contain details of a mini-course of three lectures given at the V International Course of Mathematical Analysis of Andalucía (CIDAMA), Almeria, September 12-17, 2011. When I was invited to give this mini-course and thought about possible topics for it, I decided to talk about multifunctions because they have always been present in my research on fields theoretically apart from each other as topology and integration theory. Therefore you will find here my biased views regarding part of the research that I have done over the years. The proofs for this material have been published elsewhere by me or by some other authors. This mini-survey is written attending to the invitation of the editors of this book with the sole purpose of witnessing the given mini-course and with the aim of providing the reader with connections and ideas that usually are not written in research papers. I thank the organizers of CIDAMA V as well as the editors of the book for their kind invitation to give the lecture and write this mini-survey.

In these notes we shall deal with multifunctions (or set-valued maps). Multifunctions naturally appear in analysis and topology, for instance via inequalities, performing unions or intersections with sets indexed in another set, considering the set of points minimizing an expression, etc. First, we will present some results about semi-continuity of multifunctions, namely, lower semi-continuity and an application of Michael's selection theorem. Then we will deal with upper semi-continuity of multifunctions and an application to the generation of K-analytic structures with consequences in topology and functional analysis. We will finish by showing a few results about measurability for multifunctions related to the Kuratowski-Ryll-Narzesdky selection theorem and their implications to integrability of multifunctions for non separable Banach spaces.

We describe the main interpolation methods for Banach couples, their roots and some of their properties and applications. We consider also factorization of weakly compact operators and interpolation of couples of Banach algebras.

We study optimality of function spaces that appear in Sobolev embeddings. We focus on rearrangement-invariant Banach function spaces.

This paper is an elaborated version of the material presented by the author in a three-hour minicourse at V International Course of Mathematical Analysis in Andalusia, at Almeria, Spain September 12-16, 2011. The author wishes to thank the scientific committee for the opportunity to present the course and to the organizing committee for their hospitality. The author also personally thanks Antonio Peralta for his collegiality and encouragement…

The extrapolation theorem of J. L. Rubio de Francia is one of the main results in the theory of weighted inequalities. Its proof requires a reverse factorization property and a construction of A₁ weights. We present a simple proof written in the setting of Muckenhoupt bases and discuss some extensions. In the case of the classical A_p weights the proof leads to the sharp form of the extrapolation theorem in terms of the A_p-constant of the weights.

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calderón-Zygmund operators. Namely, given 1 < p < q < ∞ and a pair of weights (u, v), if the Hardy-Littlewood maximal function satisfies the following two weight inequalities:

We review some recent results concerning game theory and their relation to some well known PDEs. In particular, we will show that solutions to certain PDEs can be obtained as limits of values of tug-of-war games when the parameter that controls the length of the possible movements goes to zero.

A well known vector measure version of the Radon-Nikodým theorem—comparing two vector measures— can also be interpreted in terms of factorization of an operator—the one associated to the first measure— through the one associated to the second one. Using several results published in recent years, we show the factorization version of this Radon-Nikodým theorem and its applications in the setting of the Operator Theory and the Harmonic Analysis.

The following sections are included:

• Introduction
• The results
• References