Abstracts for some talks (Fall Semester 2010)
Tuesday, September 7, 4:00 p.m.,
Shuanglin Shao (University of Minnesota)
On extremals to the Tomas-Stein inequality for the sphere
The adjoint Fourier restriction inequality of Tomas and Stein
asserts: \widehat{f\sigma} maps L^2(S^2) to L^4(R^3). We prove that
there exists a function which maximizes this inequality. The key
ingredient of existence is to rule out the possibility that an
extremising sequence of nonnegative functions may converge to Dirac
mass on the sphere; this is done by a concentration-compactness
argument, and a comparison with the paraboloid in the limiting case.
We also establish some exact characterizations of nonnegative
extremal functions, of complex extremals, and of complex extremising
sequences, and prove that constants are indeed local maxima.
This is joint work with Michael Christ.
Tuesday, September 14, 4:00 p.m., VV B139.
Alexander Nagel (UW Madison)
Convolution with flag distributions on homogeneous nilpotent Lie
groups.
Abstract: In joint work with Fulvio Ricci, Elias M. Stein, and Stephen Wainger,
we study singular convolution operators f -> K*f on homogeneous
nilpotent Lie groups G, where K is a special kind of tempered
distribution on G called a "flag kernel". Two of our main results
are:
A) The collection of such operators forms an algebra.
B) Each such operator is bounded on L^p(G) for 1 < p < \infty.
A major ingredient in the proofs of these theorems is a decomposition
of flag kernels into sums of dilates of normalized bump functions. In
the talk I will begin with definitions and background, and then
discuss some of the details of the decomposition.
Tuesday, September 21, 4:00 p.m., VV B139.
Andrej Zlatos (UW Madison)
Exit times of diffusions with incompressible drifts
Abstract:
We consider the influence of an incompressible drift on the expected exit
time of a diffusing particle from a bounded domain. Mixing resulting from an
incompressible drift typically enhances diffusion so one might think it
always decreases the expected exit time. Nevertheless, we show that in two
dimensions, the only simply connected domains for which the expected exit
time is maximized by zero drift are the discs.
Tuesday, October 5, 4:00 p.m., VV B139.
Zuowei Shen (National University of Singapore)
Title: Wavelet frames and applications
Abstract: One of the major driving forces in the area of applied and
computational harmonic analysis during the last two decades is the
development and the analysis of redundant systems that produce
sparse approximations for classes
of functions of interest. Such redundant systems include wavelet
frames, ridgelets, curvelets and shearlets, to name a few. This talk
focuses on tight wavelet frames that are derived from multiresolution
analysis and their applications in imaging.
The pillar of this theory is the unitary extension principle
and its various generalizations, hence we will first give a brief
survey on the development of extension principles.
The extension principles allow for systematic constructions of wavelet
frames that can tailored to, and effectively used, in
various problems in imaging science. We will discuss some of these
applications of wavelet frames. The discussion will include
frame-based image analysis and restorations, image
inpainting, image denoising, image deblurring and blind deblurring,
image decomposition, and image segmentation.
Tuesday, October 12, 4:00 p.m., VV B139.
Patrick LaVictoire (UW Madison)
Compactness of Families of Ergodic Averaging Operators
Abstract: Given a sequence of weighted ergodic averaging operators, the
existence of a pointwise ergodic theorem depends on the asymptotic
behavior of their Fourier transforms. This dependence is most apparent if
we are allowed to pass to a subsequence, in which case an ergodic theorem
can be obtained by singular integral methods. I will also discuss a
threshold between this result and a negative result, in the case of the
ergodic averages along {n^2 + \rho(n)} for a slowly growing function
\rho.
Tuesday, October 19, 4:00 p.m., VV B139.
Fedor Nazarov (UW Madison)
Sharp bounds for weighted norms of Calderon-Zygmund
operators in weighted spaces
Abstract:
Muckenhoupt and Wheeden asked if the weak $L^1$ norm of $Tf$
with respect to the weight $w$ is controlled by the $L^1$ norm
of $f$ with respect to $Mw$ where $T$ is a Calderon-Zygmund
operator, and $M$ is the Hardy-Littlewood maximal function.
Maria Carmen Reguera showed in August that it is false in the
dyadic setting
(see http://arxiv4.library.cornell.edu/PS_cache/arxiv/pdf/1008/1008.3943v1.pdf)
Now, in October, we show that even the weaker inequality
$$
\|Hf\|_{L^{1,\infty}}(w)\le \|w\|_{A_1}\|f\|_{L^1}(w)
$$
is false where $H$ is the usual Hilbert transform on the line
and $\|w\|_{A_1}=\sup\frac{Mw}w$.
This is a joint observation with A. Volberg and S. Treil.
Thursday, October 28, 2:30, VV B139.
Misha Sodin (Tel Aviv University)
Title: Uniformly spread measures and vector fields
Abstract: We show that two different notions of uniform spreading of locally
finite measures in the d-dimensional Euclidean space are equivalent. The
first notion is formulated in terms of finite distance transportations to
the Lebesgue measure, while the second notion is formulated in terms of
vector fields connecting a given measure with the Lebesgue measure.
This is a joint work with Boris Tsirelson.
Thursday, November 4, 3:30 p.m., Room TBA
Vladimir Peller (Michigan State University)
Trace formulae for perturbations of class S_m
Abstract: I am going to speak about most general trace formulae for
functions of perturbed self-adjoint operators in the case when the
perturbation belongs to the Schatten--von Neumann class $\boldsymbol{S}_m$.
The talk is based on joint work with A.B. Aleksandrov.
Tuesday, November 9, 4:00 p.m., VV B139.
Alexander Volberg (Michigan State University)
Weighted T1 theorem and solutions of A_2 conjecture
Abstract:
A_2 conjecture claimed that the norm of weighted Calder\'on--Zygmund operator
is bounded by the first degree of A_2 norm of the weight.
It turned out to be correct. We will explain
how this has been done.
Tuesday, November 16, 4:00 p.m., VV B139.
Vladimir Eiderman (visiting UW)
Singular operators with antisymmetric kernels, related capacities, and Wolff potentials.
Abstract (see also http://www.math.wisc.edu/~seeger/abstracts/eiderman.pdf)
We consider a generalization of the Riesz operator in $R^d$ and obtain
estimates for its norm and for related capacities. These estimates are based
on the certain version of T1 theorem for Calder\'on--Zygmund operators in
metric spaces. As application, we extend the known relations between
$s$-Riesz capacities, $0