Georgia Institute of Technology College of Sciences

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.

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.

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).