


Harmonic Analysis / Analyse harmonique Org: Izabella Laba (UBC) and/et Alex Iosevich (MissouriColumbia)
 MICHAEL CHRIST, University of California, Berkeley, California, USA
On multilinear oscillatory integrals

L^{p} norm inequalities are established for multilinear integral
operators of CalderónZygmund type which incorporate oscillatory
factors e^{iP}, where P is a realvalued polynomial. Our main
results concern nonsingular multilinear operators
L_{l}(f_{1},f_{2},...,f_{n}) = ò_{Rm} e^{ilP(x)} Õ_{j=1}^{n} f_{j}(p_{j}(x))h(x) dx, where
l Î R, P is a measurable realvalued function, each
f_{j} Î L^{¥}, and h Î C^{1}_{0} is compactly supported. Each
p_{j} denotes the orthogonal projection from R^{m} to a linear
subspace of R^{m} of arbitrary dimension k £ m1. Basic
questions concerning the asymptotics of such integrals as
l®¥ are posed but only partially answered. A related
problem concerning measures of sublevel sets is solved.
This is joint work with X. Li, T. Tao, and C. Thiele.
 WALTER CRAIG, McMaster University, Hamilton, Ontario L8S 4K1
On surface water waves

This talk will discuss existence and nonexistence results, as well as
results of a priori regularity, for the problem of free surface water
waves, with and without surface tension.
 GALIA DAFNI, Concordia University, Montreal, Quebec H3G 1M8
Some results on the spaces Q_{a}(R^{n})

We study the spaces Q_{a}(R^{n}), which are subpaces of BMO, and
their dyadic counterparts. We prove a quasiorthogonal decomposition
for functions in Q_{a}(R^{n}), analogous to that for functions in
BMO. This is joint work with Jie Xiao.
 ANDREA FRASER, Department of Mathematics and Statistics,
Dalhousie University, Nova Scotia B3H 3J5
A classification by multipliers of CalderónZygmund
SIO's on the Heisenberg group

We consider the class of singular integral operators on the Heisenberg
group with radial convolution kernels satisfying standard
CalderónZygmundtype conditions. In particular, this class
includes the CauchySzegö projection. Since their kernels are
radial, the operators in this class can also be described as spectral
multipliers. They form a subclass of the Marcinkiewicztype spectral
multipliers, whose convolution kernels were characterized by Müller,
Ricci, and Stein in 1995. I establish a condition on the multipliers
which characterizes this subclass.
 SINAN GUNTURK, New York University, Courant Institute, New York,
New York 10012, USA
Accuracy of onebit quantization and the fairduel problem:
some extremal problems on {1,+1} sequences

Onebit quantization refers to a class of algorithms widely used in
analogtodigital conversion to approximate bandlimited functions by
local averages of {1,+1} sequences on dense grids. The fairduel
problem (which the speaker heard from S. Konyagin) asks for a universal
ordering of shootings that makes a duel between two equal and
badshooter duelists as fair as possible, with no prior probabilistic
information. In this talk, we present the links between these problems
and report some of the recent progress. We also discuss relations to
some other extremal problems on {1,+1} sequences.
 STEVE HOFMANN, Dept. of Mathematics, University of Missouri, Columbia,
Missouri 65211, USA
L^{p} bounds for Riesz transforms

We shall give a survey of recent results establishing criteria for the
L^{p} boundedness of Riesz transforms associated to certain elliptic
operators (for example, divergence form operators on R^{n}, or the
LaplaceBeltrami operator on a complete noncompact Riemannian
manifold; we note, however, that the results under discussion are not
really about the structure of the particular operator or manifold, but
concern rather the relationship between estimates for the Riesz
transforms, and estimates for the associated heat kernels).
The starting point is an L^{2} estimate, which for the LaplaceBeltrami
operator is immediate from selfadjointness, and in the case of
divergence form operators is the recent solution of the Kato square
root problem. The L^{p} theory can therefore be viewed as the
development of some sort of CalderonZygmund machinery, in the absence
of standard regularity estimates for the singular kernels. The cases
p > 2 and p < 2 are essentially different, and the corresponding
theories have developed independently. Contributors to this subject
(sometimes jointly and sometimes independently, in various
combinations) include Auscher, Coulhon, Duong, Blunck, Kuntsmann and
Martell.
 ELLIOT KROP, Washington

 MICHAEL LACEY, Georgia Institute of Technology
Hilbert transform on smooth families of lines

This is joint work with Xiaochun Li (UCLA).
We study the operator
H_{v}f(x):=p.v. 
ó õ

1
1

f 
æ è

xyv(x) 
ö ø


dy
y



defined for smooth functions on the plane and measurable vector fields
v from the plane into the unit circle. We prove that if v has
1+e derivatives, then H_{v} extends to a bounded map from
L^{2}(R^{2}) into itself. The norm of H_{v} grows logarithmically
in the C^{1+e} norm of v.
What is noteworthy is that this result holds in the absence of some
additional geometric condition imposed upon v, and that the
smoothness condition is nearly optimal.
Whereas H_{v} is a Radon transform, for which there is an extensive
theory, our methods of proof are necessarily those associated to
Carleson's theorem on Fourier series, and the proof given by Lacey and
Thiele. These ideas can be adapted to the study o H_{v}. We find it
necessary to combine them with a crucial maximal function estimate
that is particular to the smooth vector field in question.
 MARIUS MITREA, University of Missouri, Columbia, Missouri 65211, USA
Estimates on Sobolev and Besov scales for elliptic
PDE in nonsmooth Riemannian manifolds

We study the wellposedness of the Dirichlet and Neumann problems for
the LaplaceBeltrami operator in a Lipschitz subdomain of a smooth,
compact manifold, equipped with a rough metric tensor. More
specifically, the aim is to derive sharp estimates on Sobolev and Besov
spaces when the metric tensor has a modulus of continuity satisfying a
Holder or a Dinitype condition. This is joint work with Michael
Taylor.
 GERD MOCKENHAUPT, Georgia Tech
On the Hardy Littlewood majorant property

Hardy and Littlewood observed that L^{p}spaces on the torus have the
majorant property if p is a positive even integer. For p > 2 not an
even integer it is known that the majorant property fails to hold. We
will discuss a linearized variant of the majorant problem which relates
it to local restriction problems for Fourier series with
frequency set in [0,N]. For a random selection of frequency
sets E_{w} in [0,N] of size N^{a}, 0 < a < 1, we show that for
e > 0 the events

sup
a_{n} £ 1

 
å
n Î E_{w}

a_{n} e^{2pi n x}_{p} £ C_{e} N^{e} 
å
n Î E_{w}

e^{2pi n x}_{p} 

have probability that tends to 1 as N®¥. However, for
certain frequency sets E in [0,N] we will show that above estimate
fails by a positive power in N.
 CAMIL MUSCALU, Cornell University, Department of Mathematics, Ithaca,
New York 14853, USA
On an inequality of Kato and Ponce in nonlinear PDE

We will present a generalization of the well known inequality of Kato
and Ponce in nonlinear PDE. This is recent joint work with Jill
Pipher, Terry Tao and Christoph Thiele.
 MELVYN NATHANSON, Lehman College (CUNY), Bronx, New York 10468, USA
Problems in additive number theory

This talk will be a survey recent results and unsolved problems in
additive number theory. The first part will consider sums of finite
sets of integers and lattice points, and, more generally, of finite
subsets of arbitrary abelian semigroups. Of particular interest is the
asymptotic geometric behavior of hfold sumsets as h tends to
infinity.
The second part will consider sums of infinite sets of integers and
lattice points. We consider various extremal problems of hfold
sumsets, as well as the classification of representation functions of
asymptotic bases of finite order for the integers and the nonnegative
integers.
 MALABIKA PRAMANIK, Department of Mathematics, Van Vleck Hall,
Madison, Wisconsin 53706, USA
Averaging and maximal operators for curves in R^{3}

In this work joint with Andreas Seeger, we consider the L^{p}
regularity of an averaging operator over a curve in R^{3} with
nonvanishing curvature and torsion. We also prove related local
smoothing estimates, which lead to L^{p} boundedness of a certain
maximal function associated to these averages. The common thread
underlying the proof of these results is a deep theorem of T. Wolff on
cone multipliers.
 ERIC SAWYER, McMaster

 ANDREAS SEEGER, University of WisconsinMadison
Maximal functions associated with MikhlinHoermander multipliers

We consider Fourier multipliers satisfying a condition of
MikhlinHörmander type,
fm(2^{k}·)_{Lps} <~w(k), k Î Z, 

with suitable assumptions on w(k).
We give some answers to the following question: Does the maximal function
Mf= 
sup
t > 0

F^{1}[m(t·) 
^
f

] 

define a bounded operator on L^{p}(R^{d})?
This is joint work with Loukas Grafakos and Petr Honzík.
 DANIEL TATARU, University of California, Berkeley
L_{p} bounds for the eigenfunctions of the Hermite operator

I will talk about joint work with Herbert Koch on the Hermite operator
counterpart of Sogge's L_{p} eigenfunction bounds for second order
elliptic operators.
 GUNTHER UHLMANN, University of washington, Seattle,
Washington 98195, USA
The DirichlettoNeumann map and the boundary distance
function

We will consider in this lecture the inverse boundary problem of
Electrical Impedance Tomography (EIT). This inverse method consists in
determining the electrical conductivity inside a body by making voltage
and current measurements at the boundary. The boundary information is
encoded in the DirichlettoNeumann (DN) map and the inverse problem is
to determine the coefficients of the conductivity equation (an elliptic
partial differential equation) knowing the DN map. In particular we
will consider EIT for anisotropic conductivities (the conductivity
depends on direction), which can be formulated, in dimension three or
larger, as the question of determining a Riemannian metric from the
associated DN map. We will discuss a connection of this latter problem
with the boundary rigidity problem. In this case the information is
encoded in the boundary distance function which measures the lengths of
geodesics joining points in the boundary of a compact Riemannian
manifold with boundary.

