# Rodrigo Bañuelos

- Title: Professor of Mathematics
- Research Interests: probability and its applications to harmonic analysis, partial differential equations, spectral theory, and geometry
- Office: MATH 428
- Phone: +1 765 49-41977
- Email: banuelos@purdue.edu
- Homepage: https://www.math.purdue.edu/~banuelos
- Courses: MA 53900 MA 54400

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 & Other)" 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 second term Weyl asymptotics for the number of eigenvalues in smooth, bounded,
domains in Euclidean space for the generators associated with these processes. 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), this problem remains completely open.
We should mention here that
the classical (and very powerful techniques of Ivrii, Melrose, and others) do not
apply to this problem, at least not in any direct way.
- For several years I worked on applications of martingale inequalities
to various areas of analysis and in particular to
L
^{p}-estimates for singular integrals and other operators which arise from conditional exceptions of transformations of stochastic integrals. These operators include the classical Hilbert transform, the Riesz transforms, the Beurling-Ahlfors (BA) operator, operators of Laplace trasnsform-type, versions of these for the Ornstein-Uhlenbeck semigroup and versions on manifolds. An advantage of the martingale methods is that they give sharp, or nearly sharp, bounds which often do not depend on the geometry of the space where the operators are defined. These techniques have led to the best results thus far on a celebrated conjecture of T. Iwaniec (1982) which asserts that the L^{p}-norm, for p strictly between 1 and infinity, of the BA operator is p*-1, where p* is the maximum of p and its conjugate exponent. There are 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 multipliers that can be studied by these methods can be considerably enlarged. The martingale study of these operators also leads to many interesting questions on martingale inequalities 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. Although I don't spend much time thinking about it these days
primarily because I have had no new ideas on how to proceed, I follow
with interest the work of other on the problem.
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 R
^{n}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 our article titled "On the 'hot spots' conjecture of J.Rauch," co-author with K.Burdzy of the University of Washington, Seattle, and published in 1999. I have written several other papers on this subjects which appear in my list of publications.

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