Department of Mathematics

Rodrigo Bañuelos

Rodrigo Bañuelos Portrait

RESEARCH INTERESTS
My research is at the interface of probability, harmonic analysis, and spectral theory. As reflected by my list of publications, I have devoted considerable time to the following areas of research. For a detail list of papers related to these topics, please visit "publications(MathScinet)" and "Recent Papers" by clicking on the above links.

  1. In recent years I worked on spectral and isoperimetric properties of symmetric stable and more general Lévy processes. From many points of view, these stochastic processes are natural extensions of Brownian motion. But despite the now extensive literature on their potential theory and the progress made on their spectral theory in recent years, there are many fascinating questions that remain open. We mention as an example the Weyl asymptotics for the number of eigenvalues in bounded Euclidean domains with smooth boundaries. We have advertised this problem widely in recent years and while there is some recent progress in understanding aysptotics for stable processes (such as heat trace, etc.), a second term Weyl asyptotics for the number of eigevalues involving the surface area of the domain remains open.

  2. For several years I worked on applications of martingale inequalities to various areas of anlsysis and in particular to Lp-estimates for singular integrals and other operators which arise from conditional expections of trasformations of stochastic integrals. These operartors include the classical Hilbert transform, the Riesz transforms, the Beurling-Ahlfors (BA) operator, and operators of Laplace trasnsform-type. A celebrated conjecture of T. Iwaniec (1982) asserts that the Lp-norm, for p stricly between 1 and infinity, of the BA operator is p*-1, where p* is the maximum of p and its conjugate exponent, and this has been the motivation for some of this work. There are many interesting consequences of this conjecture to quasiconformal mappings and to regularity results for solutions of certain nonlinear PDE's. By replacing the Brownian motion with other Lévy processes, the class of multiplers that can be studied by these methods can be considreably enlarged. The martingale study of these operators also leads to many interesting questions on martingales inequalties and other applications and these have been investigated in recent years by several authors.

  3. The "hot-spots" conjecture has been one of my favorite problems for several years. The conjecture, made in 1974 by Jeff Rauch of the University of Michigan (which I learned directly from Jeff during a visit to Ann Arbor in the mid 90's) asserts that the maximum (and minimum) of the first non-constant Neumann eigenfunction for a smooth bounded domain in Rn is attained on the boundary and only on the boundary of the domain. Without any assumptions on the domain the conjecture is false. However, it is widely believe to be true for convex domains but this remains open even in two dimensions and even for arbitrary triangles. With some understanding of the geometry of the nodal curve of the eigenfunction, the conjecture reduces to a maximum principle for a mixed (Dirichlet-Neumann) boundary value problem. One can then relate this problem to properties of Brownian motion in the domain with killing and reflection. The Brownian motion approach to the problem was initiated in the article title "On the 'hot spots' conjecture of J.Rauch," co-author with K.Burdzy of the University of Washington, Seattle, and published in 1999.