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Series: Analysis Seminar

We shall describe how the study of certain measures called
reflectionless measures can be used to understand the behaviour of
oscillatory singular integral operators in terms of non-oscillatory
quantities. The results described are joint work with Fedor Nazarov,
Maria Carmen Reguera, and Xavier Tolsa

Series: Analysis Seminar

We consider generalized Bochner-Riesz multipliers $(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. For integers $m\ge 2$, we find domains such that $(1-\rho(\xi))_+^{\lambda}\in M^p(\mathbb{R}^2)$ for all $\lambda>0$ in the range $\frac{m}{m-1}\le p\le 2$, but for which $\inf\{\lambda:\,(1-\rho)_+^{\lambda}\in M_p\}>0$ when $p<\frac{m}{m-1}$. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the ``additive energy" of the boundary of a convex domain. Second, we associate a set of directions to a convex domain and define a sequence of Nikodym-type maximal operators corresponding to this set of directions. We show that domains that have low higher order energy, as well as those which have asymptotically good $L^p$ bounds for the corresponding sequence of Nikodym-type maximal operators, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.

Series: Analysis Seminar

In my talk, I will discuss coordinate shifts acting on Dirichlet spaces on the bidisk and the problem of finding cyclic vectors for these operators. For polynomials in two complex variables, I will describe a complete characterization given in terms of size and nature of zero sets in the distinguished boundary.

Series: Analysis Seminar

The purpose of this talk is to introduce some recent works on
the field of Sobolev orthogonal polynomials. I will mainly focus on
our two last works on this topic. The first has to do with orthogonal
polynomials on product domains. The main result shows how an orthogonal
basis for such an inner product can be constructed for certain weight functions,
in particular, for product Laguerre and product Gegenbauer weight functions.
The second one analyzes a family of mutually
orthogonal polynomials on the unit ball with respect to an inner
product which involves the outward normal derivatives on the sphere.
Using the representation of these polynomials in terms of spherical
harmonics, algebraic and analytic properties will be deduced. First,
we will get connection formulas relating classical multivariate
orthogonal polynomials on the ball with our family of Sobolev
orthogonal polynomials. Then explicit expressions for the norms will
be obtained, among other properties.

Series: Analysis Seminar

Examples of analytic fractals are Julia sets, Koch Curves, and Sierpinski
triangles, and graphs of analytic functions. Given a piece of such a set, how does one
"continue" it, in a manner consistent with the classical construction of an analytic
Riemannian manifold, starting from a locally convergent series expansion?

Series: Analysis Seminar

In the recent past multiple orthogonal polynomials have attracted great attention.
They appear in simultaneous rational approximation, simultaneous quadrature rules,
number theory, and more recently in the study of certain random matrix models.
These are sequences of polynomials which share orthogonality conditions with respect
to a system of measures. A central role in the development of this theory is played
by the so called Nikishin systems of measures for which many results of the standard
theory of orthogonal polynomials has been extended. In this regard, we present some
results on the convergence of type I and type II Hermite-Pade approximation for a
class of meromorphic functions obtained by adding vector rational functions with real
coefficients to a Nikishin system of functions (the Cauchy transforms of a Nikishin
system of measures).

Series: Analysis Seminar

We will prove a recent version of the weighted Carleson Embedding Theorem
for vector-valued function spaces with matrix weights. Time permitting, we
will discuss the applications of this theorem to estimates on
well-localized operators. This result relies heavily on the work of Kelly
Bickel and Brett Wick and is joint with Sergei Treil.

Series: Analysis Seminar

We will prove a pointwise estimate for positive dyadic shifts of complexitym which is linear in the complexity. This can be used to give a pointwiseestimate for Calderon-Zygmund operators and to answer a question posed byA. Lerner. Several applications will be discussed.- This is joint work with Jose M. Conde-Alonso.

Series: Analysis Seminar

The classical Rubio de Francia extrapolation allows you to obtain strong-type estimates for weights in A_p (and every p>1) if you can show that it holds for some p_0>1. However, the endpoint p=1 has to be treated separately. In this talk we will explain how to deduce weak-type (1,1) estimates for A_1 weights if we have a certain restricted weak-type inequality at some level p_0>1. We will then show how this approach can be applied to the Bochner-Riesz operator at the critical index and Fourier multipliers.

Series: Analysis Seminar

The conventional point of view is that the Lagrangian is a scalar
object, which through the Euler-Lagrange equations provides us with one
single equation. However, there is a key integrability property of certain
discrete systems called multidimensional consistency, which implies that we
are dealing with infinite hierarchies of compatible equations. Wanting this
property to be reflected in the Lagrangian formulation, we arrive naturally
at the construction of Lagrangian multiforms, i.e., Lagrangians which are
the components of a form and satisfy a closure relation. Then we can
propose a new variational principle for discrete integrable systems which
brings in the geometry of the space of independent variables, and from this
principle derive any equation in the hierarchy.