# Abstracts

**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 to . 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.