


Harmonic Analysis
Org: Izabella Laba (UBC) and Malabika Pramanik (Caltech; UBC) [PDF]
 CHRISTINE CARRACINO, Richard Stockton College of NJ, Pomona, NJ 08240
Estimates for the Szegö kernel and the [`(¶)]_{b}
equation
[PDF] 
We can consider the Szegö kernel S(z,z) on the boundary of
model domains W = {(z_{1},z_{2}) Î C^{2}   Áz_{2} > b(Âz_{1})}. If b is convex, the only singularities of
S(z,z) are on the diagonal z=z. When the function b is
a certain nonconvex function, we show that near certain points, there
are singularities off the diagonal.
We will then discuss some recent work on a related problem, the
[`(¶)]_{b} problem for the model domain considered above.
This is joint work with Jennifer Halfpap.
 JIM COLLIANDER, University of Toronto
Strichartz refinements and concentration in NLS
[PDF] 
Refinements of the Fourier restriction estimate have been used by
Bourgain and others to prove mass concentration occurs in finite time
blowup solutions of mass critical nonlinear Schrodinger equations.
I'll revisit this argument and remark on certain quantifications and
extensions.
 GALIA DAFNI, Department of Mathematics and Statistics, Concordia
University, 1455 de Maisonneuve Blvd. West, Montreal,
Quebec, H3G 1M8, Canada
Classes of measures generated by capacities
[PDF] 
For b < 1, the fractional Carleson condition for a measure m
on the upper halfspace R^{n+1}_{+}:
for all balls B in R^{n}, where B is Lebesgue measure
and T(B) is the tent over B, is not equivalent to the same
condition for open sets. In joint work with Georgi Karadzhov and Jie
Xiao, we show that these Carlesontype conditions (involving balls)
are equivalent to conditions bounding the measure of the tent T(O)
over an open set O by a function of its capacity. The capacities
used include Riesz, Bessel, Besov and Hausdorff capacities. These
conditions are analogous to conditions introduced by Maz'ya for
measures on R^{n}.
 BURAK ERDOGAN, University of Illinois at UrbanaChampaign
Strichartz estimates for Schrodinger equations with large
magnetic potentials
[PDF] 
We show that the time evolution of the magnetic Schrodinger operator
in R^{3} satisfies global Strichartz and smoothing estimates
under suitable smoothness and decay assumptions on A and V but
without any smallness assumptions. We require that zero energy is
neither an eigenvalue nor a resonance.
This is a joint work with M. Goldberg and W. Schlag.
 RALUCA FELEA, Rochester Institute of Technology
Composition calculus in inverse scattering
[PDF] 
Fourier integral operators (FIOs) with singularities are used in
inverse scattering theory. In such problems, artifacts can appear
when inversion is attempted and one would like to understand precisely
and eliminate them as far as possible. The operators considered here
arise in linearized seismic imaging and synthetic aperture radar. The
singularities which appear in this problems are folds, submersion with
folds and cross caps. One would like to understand the composition of
such operators since in general, the composition of two FIOs is not a
FIO. We will establish a composition calculus for FIOs associated to
folding canonical relations, working away from the fold points.
 MICHAEL GREENBLATT, University at Buffalo
A resolution of singularities and some applications
[PDF] 
A geometric resolution of singularities algorithm for realanalytic
functions is described. This method is elementary in its statement
and proof, extensively using explicit coordinate systems. Each
coordinate change used in the resolution procedure is onetoone on
its domain, and is of one of a few explicit canonical forms.
Applications to classical analysis are given, including a theorem
regarding the existence of critical integrability exponents.
 ALLAN GREENLEAF, University of Rochester, Department of Mathematics,
Rochester, NY 14627, USA
Counterexamples to Calderón's problem as the mathematics
of invisibility
[PDF] 
We show how earlier work on counterexamples for the Calderón problem
(i.e., whether a conductivity function or tensor s(x)
on a domain W is determined by the boundary values of the
solutions to Ñ·(sÑ)u = 0) can be extended to
obtain rigorous results concerning invisibility (or "cloaking") for
solutions of the Helmholtz and Maxwell equations.
This is joint work with Yaroslav Kurylev, Matti Lassas and Gunther
Uhlmann.
 KATHRYN HARE, Department of Pure Mathematics, University of Waterloo,
200 University Ave. W., Waterloo, Ontario, N2L 3G1
Directional maximal operators with smooth densities
[PDF] 
We investigate the mapping properties of directional maximal operators
on the plane with smooth densities. If vectors of all lengths in a
given set of directions are taken in defining the maximal operator,
then the boundedness of the operator depends on the order of the
stationary points of the density function. In contrast, if the set of
directions is limited to sums of diadic directions, and the density
function has only finitely many stationary points, each of finite
order, the maximal operator is bounded on all L^{p} for p > 1.
This is joint work with M. Roginskaya.
 ALEX IOSEVICH, MissouriColumbia

 AHYOUNG KIM, University of WisconsinMadison
Discrete Schrodinger Operators with Slowly Oscilating
Potentials
[PDF] 
We investigate asyptotic behavior of the generalized eigenfunctions of
discrete Schrodinger operator with potentials ¶b(n) Î l^{p}(Z), 1 < p < 2. Our approach is based on the work of Christ and
Kiselev in the continuous case. The main new step is a development of
discrete version of multilinear operator analysis and maximal function
estimate.
Joint work with A. Kiselev.
 SUNG EUN KIM, University of Rochester, Rochester, NY
Calderon's inverse problem for piecewise linear
conductivities
[PDF] 
We will show the global uniqueness for the Calderon's inverse problem
for a piecewise linear, or more generally piecewise smooth across
polyhedra, conductivity in R^{3} by using the geometry of Faddeev
Green's function.
 MICHAEL LACEY, Geogia Tech
On the Small Ball Problem
[PDF] 
We consider Haar functions in the unit cube in three dimensions,
normalized in L^{¥}. The question at hand is a `nontrivial'
lower bound on the L^{¥} norm of the sum

å
R=2^{n}

a_{R} h_{R} (x). 

The key point of the sum is that is formed over rectangles of a fixed
volumethis is the `Hyperbolic' assumption. We prove that for some
h > 0, we have the estimate
 
å
R=2^{n}

a_{R} h_{R} (x) _{¥} > cn^{1+h}2^{n} 
å
R=2^{n}

a_{R} 

(h = 0 is the `trivial' estimate). In a prior result of J. Beck, a
famous and famously difficult result, established a logarithmic gain
over the trivial estimate. We simplify and extend Beck's argument to
prove this result.
Joint work with Dmitry Bilyk.
 NEIL LYALL, University of Georgia
On a Theorem of Sárközy and Furstenberg
[PDF] 
A theorem of Sárközy and Furstenberg states that if A Í Z with positive upper Banach density, then the set of
quadratic return times {d  AÇ(A+d^{2}) ¹ Æ} is
nonempty. Using Fourier analysis we give a new proof of the fact
that the set of all quadratic return times is syndetic and obtain
uniform lower bounds for the density of these return times.
This is joint work with Ákos Magyar.
 AKOS MAGYAR, University of Georgia
Maximal operators associated to discrete subgroups of
nilpotent Lie groups
[PDF] 
A maximal theorem is proved for averages taken over suitable discrete
subvarieties of nilpotent Lie groups.
 ADRIAN NACHMAN, University of Toronto
Imaging Obstacles In Inhomogeneous Media
[PDF] 
We show that an obstacle inside a known inhomogeneous medium can be
determined from the scattering amplitude at one frequency. Moreover,
we show that if the scattering operators for two obstacles are
comparable at one nonresonant frequency, then the obstacles must
coincide. The proof gives a new reconstruction procedure; it is based
on an extension of the factorization method, which I'll review.
This is joint work with Lassi Paivarinta and Ari Teirila.
 ALEXANDER NAGEL, Department of Mathematics, University of Wisconsin,
480 Lincoln Drive, Madison, WI 53706, USA
Bergman and Szego kernels on tubular domains near points of
infinite type
[PDF] 
In joint work with Jennifer Halfpap and Stephen Wainger, we obtain
estimates for the Bergman and Szegö kernels in domains of the form
W = 
ì í
î

(z_{1}, z_{2}) Î C^{2} 
ê ê

Á[z_{2}] > b (Â[z_{1}]) 
ü ý
þ



where b Î C^{¥} (R) is convex and even, with
b"(r) > 0 for r > 0 and b^{(n)}(0) = 0 for all nonnegative
integers n. For example, if b_{a}(r) = exp(r^{a})
for r small, we show that the Bergman and Szegö kernels have
singularities away from the diagonal of the boundary of W if
and only if a ³ 1.
 RICHARD OBERLIN, University of WisconsinMadison, 480 Lincoln Dr., Madison,
WI 53706
The (d,k) Kakeya problem
[PDF] 
A (d,k) set is a subset of R^{d} containing a translate of
every kdimensional plane. We use mixednorm estimates for the
xray transform to obtain improved dimension estimates for (d,k)
sets.
 JILL PIPHER, Brown University
Square function variants adapted to boundary value problems
[PDF] 
A variant of the square function has applications to boundary value
problems and to characterizing A^{¥} weights.
 ERIC SAWYER, McMaster University, Hamilton, Ontario
Interpolating sequences for the Dirichlet space
[PDF] 
In the 1950's Buck raised the question of whether or not there exists
an infinite subset Z = {z_{j}}_{j=1}^{¥} of D that is
interpolating for H^{¥}(D). In 1958 Carleson gave an
affirmative answer and characterized all such interpolating sequences
in the disk. In 1961 Shapiro and Shields demonstrated the equivalence
of this problem with certain Hilbert space analogues involving
l^{2}(m) where m = å_{j} k_{zj}(z_{j})^{1}d_{zj} and k_{z} is the reproducing kernel. Suppose that Z = {z_{j}}_{j=1}^{¥} Ì D is separated and let R be the
restriction map Rf = {f(z_{j})}_{j=1}^{¥}. Then the results
of Carleson and Shapiro and Shields show the following four conditions
are equivalent: R maps H^{¥}(D) onto l^{¥}(Z); R
maps H^{2}(D) onto l^{2}(m); R maps H^{2}(D) into
l^{2}(m); m( T(I) ) £ CI for all arcs I Ì T. In 1995 Marshall and Sundberg extended this theorem to the
Dirichlet space D(D) and its multiplier algebra M_{D(D)}, but
without the second condition. In fact, as observed by Bishop in 1995,
even when the measure m is finite, R maps D(D) onto
l^{2}(m) for sequences Z more general than those for which R
maps D(D) into l^{2}(m). We give two results toward resolving
the open question of when R maps D(D) onto l^{2}(m): the
first is a geometric characterization of such Z in the case m is
a finite measure, and the second shows there are such Z with m
an infinite measure, thus answering a question of Bishop.
 ANDREAS SEEGER, University of WisconsinMadison
Fourier restriction theorems for curves
[PDF] 
We shall report recent joint work with JongGuk Bak and Daniel Oberlin
on Fourier restriction for curves in R^{d} and related oscillatory
integral operators.
 ROMAN SHTERENBERG, University of WisconsinMadison, Math. Dept., 480 Lincoln
Dr., Madison, WI 537061388, USA
Blow up and regularity for Burgers equation with fractional
dissipation
[PDF] 
We present a comprehensive study of the existence, blow up and
regularity properties of solutions of the Burgers equation with
fractional dissipation. We prove existence of the finite time blow up
for the power of Laplacian a < 1/2, and global existence as
well as analyticity of solution for a ³ 1/2. We also
discuss solutions with very rough initial data.
 GIGLIOLA STAFILLANI, MIT, Room 2246, 77 Massachusetts Ave., Cambridge, MA 02139
On the L^{2} critical NLS
[PDF] 
In this talk I will give an overview of the global wellposedness
results for the L^{2} critical NLS below the H^{1} norm. The recent
result of TaoVisanZhang proves global wellposedness in L^{2} for
radial data in higher dimensions d ³ 3. This problem is still
open in lower dimensions and for nonradial data. In this talk I will
present some recent results of global wellposedness that I obtained
with De Silva, Pavlovic and Tzirakis for any data in H^{sd} for a
certain 0 < s_{d} < 1.
 PAUL TAYLOR, Shippensburg University, 1871 Old Main Dr., Shippensburg,
PA 17011, USA
BochnerRiesz Means With Respect to a 2 by 2 Cylinder
[PDF] 
We present estimates for BochnerRiesz means with respect to the
R^{2} ×R^{2} cylinder variant { x, max{x_{1}^{2} + x_{2}^{2}, x_{3}^{2} + x_{4}^{2}} = 1 }. The operator
S^{l} to be studied is given by

^
S^{l}f

(x) = 
æ è

1  
max
 
ì í
î

 Ö

x_{1}^{2} + x_{2}^{2}

,  Ö

x_{3}^{2} + x_{4}^{2}


ü ý
þ


ö ø

l +

. 

 ERIN TERWILLEGER, University of Connecticut
The WienerWintner Theorem for the Hilbert Transform
[PDF] 
Our main result is an oscillation inequality which is an extension of
Carleson's Theorem on Fourier series. As a consequence one obtains
the following extension of the WienerWintner Theorem on ergodic
averages: for all measure preserving flows (X,m,T_{t}) and f Î L^{p}(X,m), there is a set X_{f} Ì X of probability one, so that
for all x Î X_{f} we have

lim
s¯ 0


ó õ

s < t < 1/s

e^{iqt} f(T_{t} x) 
dt
t

exists for all q. 

This work is joint with Michael Lacey.

