Rodrigo Bañuelos
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- Indiana Univ. Math. J.(RSS)
- Analogs of principal series representations for Thompson's groups $F$ and $T$
- Blow-up set for a semilinear heat equation and pointedness of the initial data
- Canonical transfer-function realization for Schur multipliers on the Drury-Arveson space and models for commuting row contractions
- Comparison results for capacity
- Curvature-adapted submanifolds of symmetric spaces
- Degree growth of monomial maps and McMullen's polytope algebra
- Title: Professor of Mathematics
- Office: MATH 428
- Phone: 765-494-1977
- Email: banuelos@math.purdue.edu
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.
- 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.
- 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.
- 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.