Advanced Courses of Mathematical Analysis VI

The following sections are included:

• Preface
• Organizing committees
• Contents

The main idea of these notes is to present the Kannan-Lovász-Simonovits spectral gap conjecture on the correct estimate for the spectral gap of the Laplace-Beltrami operator associated to any log-concave probability on ℝⁿ.

The following sections are included:

• Introduction
• Preliminaries
• Dyadic operators
• Digression: one weight inequalities
• Testing conditions
• Bump conditions
• Separated bump conditions
• References

The topic of this short survey is composition operators, i.e., maps of the shape ƒ → ƒ˚φ, where the symbol φ: D → D is holomorphic. We shall give a (non-exhaustive) overview of -more or less recent- results on such operators, acting on Hardy spaces. The story involves some classical tools of complex analysis, as Nevanlinna counting function and Carleson measures. We shall illustrate this presentation with miscellaneous examples and questions. We shall pay attention to their possible membership to various classes of operator ideals, and in particular, in themembership of the class of absolutely summing operators.

The main purpose of this survey is to gather results on the boundedness of the Bergman projection. First, we shall go over some equivalent norms on weighted Bergman spaces which are useful in the study of this question. In particular, we shall focus on a decomposition normtheorem for radial weights ω with the doubling property .

The meanings given to the terms “algebra” and “analysis”, separately or jointly in expressions like “algebraic analysis”, have changed over time, even simultaneously being used with significant differentiating shades.

I will focus on the period between the publication, from 1751 on, of the Encyclopédie edited by Diderot and D’Alembert, and the publication from 1899 on, of the Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, driven by F. Klein. The latter was translated into French with slight additions, under the direction of J.Molk, from 1904 until 1915, when the process was interrupted by the Great War (1914-1918), which also marks the end of the period covered in this paper.

This survey is a slightly extended version of the lecture given by the author at the VI International Course of Mathematical Analysis in Andalucía (CIDAMA), in September 2014. Most results formpart of the paper [3], written jointly with S. Hartzstein, T. Signes, J.L. Torrea and B. Viviani.

We report on recent advances on one of the central notions in linear dynamics, that of a frequently hypercyclic operator. We include a list of ten open problems.

General aspects of the domination and factorization properties of multilinear operators from quasi-Banach lattices to quasi-Banach spaces are presented.

Using a modern approach that allows a full generality, we analyze some classical and new results, showing when certain vector valued norm inequalities provide factorization through some well-known Banach function lattices, as for instance Orlicz spaces. For example, we show that our framework allows to prove a multilinear variant of the Grothendieck factorization theoremfor products of C(K)-spaces.

This article reviews the recent developments on the boudedness of products of Toeplitz operators. Special attention is paid to the solution of a conjecture posted by D. Sarason in the 90’s on the product of Toeplitz operators in the Bergman space recently achieved by Aleman, Pott and the author in [1].

In this note we introduce the semigroup language as a natural tool to develop a Harmonic Analysis associated to general Laplacians. We make a quick walk through the classical cases of the torus and the line in order to motivate the introduction of semigroups. The case of the discrete Laplacian in the integers is considered in more detail.

The following section is included:

• Author index