Tuesday 2-3 PM, MSB 111 

Abstract: Croft, Falconer and Guy posed a series of questions generalizing Ulam's floating body problem, as follows.

Given a convex body K in R^3, we consider its plane sections with certain given properties,

   (V): Plane sections which cut off a given constant volume 

1. Plane sections which have a given constant area

   (I) Plane sections which have equal constant principal moments of inertia

It has been known for quite some time that various conditions on the oscillation of the matrix $A$ in elliptic/parabolic operators of the form 

 $L = -div A \nabla$ or $L = \partial_t - \div A \nabla$ 

An exponential basis on a measurable domain of $\Bbb{R}^d$ is a Riesz basis in the form of

We prove a wavelet $T(1)$ theorem for compactness of multilinear Calder\'{o}n-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.

Extending the Sobolev theory of quasiconformal and quasiregular maps to subdomains of the complex plane motivates our investigation of Sobolev regularity of singular integral operators on domains. We introduce new paraproducts which lead to higher order T1-type testing conditions. A special case provides weighted Sobolev estimates for the compressed Beurling transform which imply quantitative Sobolev estimates for the Beltrami resolvent. This is joint work with Francesco Di Plinio and Brett D. Wick.

We discuss various results on singular integrals adapted to entangled dilations from the past two years. The existing results are mostly on the so-called Zygmund dilations that constitute the simplest intermediate dilation structure lying in between the classical one-parameter setting and the multi-parameter setting.

We consider an elliptic operator, double divergence form operator L*, which is the formal adjoint of the elliptic operator in non-divergence from L. An important example of a double divergence form equation is the stationary Kolmogorov equation for invariant measures of a diffusion process. We are concerned with the regularity of weak solutions of L*u=0.

We will also discuss some applications and parabolic counterparts.

In the 1960's, T. Kato posed a conjecture about finding the domain and some crucial estimates for the square root of elliptic partial differential operators in divergence form. The question attracted lots of interest because of the applications that it would have, and it turned out to be fairly tough to prove: only after around 40 years and joint efforts from different areas in Analysis (mainly PDE, Functional Analysis and Harmonic Analysis), it was finally solved by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in 2002.