Abstract: We present some recent results in the theory of Fourier multipliers. These concern sharp versions of the classical multiplier theorems of Hörmander and Marcinkiewicz and their bilinear analogues. We also discuss optimal, and in some cases necessary and sufficient, criteria for certain bilinear Fourier multiplier operators to be bounded from L^2 x L^2 to L^1. The lectures cover work of the speaker and/or of Danqing He, Petr Honzík, Hanh Van Nguyen, and Lenka Slavíková.

Abstract: In these series of lectures we will study the Dirichlet problem in the upper half-space for second-order, homogeneous, elliptic systems, with constant complex coefficients, such as the Laplacian or the Lamé system of elasticity. Using techniques from Harmomic Analysis and Partial Differential Equations we will present the main ideas which allow us to establish the well-posedness of the Dirichlet problem with data in Lebesgue spaces, Köte function spaces, weighted Lebesgue spaces, BMO, VMO, etc. By the seminal work of S. Agmon, A. Douglis, and L. Nirenberg, there exists a Poisson kernel associated with each of the previous operators which is an object whose properties mirror the most basic characteristics of the classical harmonic Poisson kernel. This in turn can be used to construct solutions associated with any locally integrable function satisfying some mild decay condition at infinity. The study of uniqueness is however more delicate and requires the use of the properties of the elliptic operators, along with the nature of the the spaces of solutions we are considering. Related to the latter we will also present some Fatou type results guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems, allowing us to recover the null-solutions from their nontangential boundary traces. For the sake of simplicity, most of the course will just treat the model case of the Laplace operator, but we will use modern techniques that can be employed to consider the general case of systems with constant complex coefficients. The content of this course is part of a joint project with D. Mitrea, I. Mitrea, and M. Mitrea.

Abstract: The isoperimetric problem in a Euclidean space with density has obtained a lot of attention in the last decades. The goal is the same as usual, that is, to find a set minimizing the perimeter among those with a fixed volume. However, "perimeter" and "volume" are now calculated with respect to two given functions, usually called "densities". In a complete generality, no existence or regularity whatsoever is to be expected; nevertheless, reasonable assumptions on the densities ensure good properties. In this course, we will present the main history of the problem, with several examples, and we will see some of the newest results on the subject, some of which still unpublished.