


Harmonic Analysis / Analyse harmonique Org: Galia Dafni (Concordia)
 JAMES COLLIANDER, University of Toronto, 100 St. George Street, Toronto, ON
M5S 3G3
On rough blowup solutions of L^{2} critical NLS

The goal of the work reported upon in this talk is to understand the
L^{2} critical nonlinear Schrödinger (NLS) blowup dynamics for rough
initial data. Much of the known theory for NLS blowup relies upon
energy conservation and is thus restricted to H^{1} solutions, even
though the NLS evolution problem is wellposed in L^{2}. Two new
results will be described:
(1) For all s satisfying s_{Q} < s < 1, H^{s} blowup
solutions concentrate at least the mass of the ground state soliton
into a point at blowup time.
(2) L^{2} blowup solutions concentrate at least a fixed amount
(conjectured to be the mass of the ground state) of L^{2} mass at
blowup time.
 PAUL GAUTHIER, Université de Montréal, Centreville, Montréal, QC
H3C 3J7
Completeness of certain function spaces

We recall the maximum principle on unbounded domains and discuss some
recent applications thereof to determine when certain spaces of
harmonic and holomorphic functions are complete.
 PENGFEI GUAN, McGill University
Homogeneous Weingarten curvature equations

If M is a starshaped hypersurface about the origin in R^{n+1}, by dilation property of the curvature function, the
kth Weingarten curvature can be considered as a function of
homogeneous degree k in R^{n+1} \{0}. The
homogeneous Weingarten curvature problem is: given a homogeneous
function F of degree k in R^{n+1} \{0}, does
there exist a starshaped hypersurface M such that its kth
Weingarten curvature is at x Î M is equal to F(x)? The problem
is a nonlinear eigenvalue problem for the curvature equation. When
k=n, the resultant equation is a MongeAmpère type, the problem
was studied by Delanoë; on the other end k=1, the equation is
of mean curvature type which was studied by Treibergs. We will
describe a recent joint work with X. Ma and C. S. Lin on the problem
for all cases 1 £ k £ n. We also discuss the existence of
convex solutions.
 KATHRYN HARE, Department of Pure Mathematics, University of Waterloo,
Waterloo, ON N2L 3G1
Energy and Hausdorff dimensions of measures

Energy and Hausdorff dimension are two related ways to quantify the
singularity of a measure. There is a classical formula relating the
energy of a positive measure on R^{n} with a weighted L^{2} norm of
its Fourier transform. We discuss an extension of this formula to
measures on compact, connected, Riemannian manifolds, replacing the
Fourier transform with the projections of the measures onto the
eigenspaces of the Laplacian operator. We also discuss extending the
definition of energy to signed measures and show that this generalized
energy dimension continues to provide a lower bound for the Hausdorff
dimension of the measure.
 XINHOU HUA, University of Ottawa
Meromorphic Functions Sharing the Same Zeros and Poles

In this paper, we solved Hinkkanen's problem proposed in 1984: Any
meromorphic function f is determined by its zeros and poles and the
zeros of its first four derivatives. We also show that the number 4
is best possible by a counterexample.
 MARIANNE KORTEN, Kansas State University, Department of Mathematics,
138 Cardwell Hall, Manhattan, KS 66502
Existence, uniqueness and regularity of the free boundary in
the HeleShaw problem

We obtain the unique weak solution (u^{¥},V) to the HeleShaw
problem with a mushy zone 0 £ u^{¥} £ 1

u^{¥}_{t} = DV(x,t) Î G^{c} ×(0,+¥), 
 
  
 V(x,t) = p(x), (x,t) Î ¶G ×(0,+¥). 
  (1) 
This model describes the flow of a viscous fluid being injected
through the slot G between two nearby plates, and is used in
injection molding for the production of packaging materials and the
interior plastic parts of cars and airplanes, in electromechanical
machining, and to study the diffusion of nutrients and medicines
within certain tumors.
We find u^{¥}, and V as the (pointwise) "Mesa" type limit
of u^{(m)} and u ( u_{m}, m(u_{m}1)_{+} ), where the u_{m}
are solutions to onephase Stefan problems with increasing
diffusivities u^{(m)}_{t} = D(u^{(m)}1)_{+}, with fixed initial
and boundary data 0 £ u_{I} £ 1 and p(x) > 0. The main tools
are information available from Korten's earlier work about solutions
and interphases in the onephase Stefan problem, Blank's results on
the obstacle problem, and new energy estimates for Stefan problems due
to Moore and Korten.
We
obtain the traditional formulation (by means of the Baiocchi
transformation) as an obstacle problem. At this point regularity in
space follows from the work of Blank and Caffarelli.
Joint work with I. Blank and C. N. Moore.
 ISABELLA LABA, University of British Columbia, Vancouver, BC
Wolff's inequality for hypersurfaces

Wolff (2000) proved an L^{p} "local smoothing" inequality for
circular cones in dimension 3 and for large enough p, and used it to
obtain sharp L^{p} estimates for cone multipliers and local smoothing
bounds for solutions of the wave equation. This result was extended
to higher dimensions by Wolff and myself. In a joint work with
Malabika Pramanik, we extend it further to a wider class of
hypersurfaces of codimension 1, including generic surfaces with
negative curvatures and certain surfaces with more than one flat
direction.
 AKOS MAGYAR, University of Georgia
A note on Fourier restriction and the Newton polygon

Local L^{p}  L^{2} bounds are proved for the restriction of the Fourier
transform to analytic surfaces of the form: S = ( x,f(x) )
in R^{3}. It is found that the range of exponents are determined by
the socalled distance of the Newton polygon, associated to f,
except when the principal part of f(x) contains a factor of high
multiplicity. The proofs are based on the method of PhongStein and
Rychkov, adapted to scalar oscillatory integrals.
 JAVAD MASHREGHI, Université Laval
One Multiplier theorem, several proofs

A model subspace of the Hardy space H^{2} generated by the inner
function Q is K_{Q} = H^{2}  Q H^{2}. The model
subspaces generated by Q = e^{i sQ}, s > 0
have been extensively studied. The classical mutiplier theorem of
BeurlingMalliavin characterizes the admissible majorants of these
special and very important model subspaces of H^{2}. The original
proof is rather difficult. Thereafter, many mathematicians, including
in particular Paul Koosis, worked in this direction and some gave new
proofs of the classical theorem. Victor Havin, Fedya Nazarov and I
also gave a new proof. We believe that our proof is the simplest one
and besides our main theorem also works for other model subspaces
of H^{2}.
 MING MEI, Concordia University, Montreal, Canada
Phase transitions to 2×2 system of conservation laws
with periodic boundary condition

The study focuses on the phase transitions of a 2×2 psystem
of viscositycapillarity with periodic initialboundary condition in
viscoelastic material. The goal of the present study is to make a
novel contribution to the open problem: the asymptotic behavior of the
solution of this model. The location of the initial data and the
amplitude of viscosity play a key role for the phase transitions. The
criteria of the phase transition solutions are provided for both the
original problem and its steadystate periodic boundary value problem.
Furthermore, the convergence rate of the steadystate solutions is
obtained. A crucial step in this study is the proof of the uniform
boundedness of the solution by Liapunov functional. Finally,
numerical simulations are carried out to confirm the theoretical
results.
This is a joint work with Yau Shu Wong at University of Alberta and
Liping Liu at Duke University.
 MARIUS MITREA, University of Missouri
The solution of the ChangKrantzStein conjecture

In this talk I will report on some recent joint work with S. Mayboroda
on optimal regularity results for the harmonic Green potential G on
Besov and TriebelLizorkin scales in Lipschitz domains. As is
wellknown, the latter class contains, as a particular case, the Hardy
space H^{p}. When specialized to this setting, our results prove that
two derivatives on G map H^{p} to itself for a small interval of
values p < 1. For Lipschitz domains, this has been conjectured to be
the case by ChangKrantzStein in the early 1990s.
 CAMIL MUSCALU, Cornell University
Paraproducts on polydiscs

Last year, at the AMS/CMS Meeting in Vancouver, we presented a result
which extended the classical CoifmanMeyer theorem to the biparameter
setting of the bidisc. We also pointed out that the dparameter
case is open, for d > 2. This open case has been resolved in the
meantime and the corresponding theorem will be described during our
talk.
This is recent joint work with Jill Pipher, Terry Tao and Christoph Thiele.
 PHILIPPE POULIN, McGill University, 845 rue Sherbrooke Ouest, Montreal,
H3A 2T5
The MolchanovVainberg Laplacian

It is well known that the Green function of the standard discrete
Laplacian on a lattice exhibits a pathological behavior in dimension
greater than 2. Molchanov and Vainberg suggested an alternative to
the usual Laplacian and conjectured that a polynomial decay holds for
this latter. This talk presents a proof of this conjecture.
 TOM RANSFORD, Université Laval, Dép. de mathématiques, Québec
(QC), G1K 7P4
Invariant subspaces of the Dirichlet space

One of the cornerstones of the theory of Hardy spaces is Beurling's
invariantsubspace theorem, which classifies the closed subspaces
invariant under the shift operator. The corresponding classification
in the Dirichlet space is still an open problem. I will discuss some
recent progress in this area.
Joint work with Omar ElFallah and Karim Kellay.
 ERIC SAWYER, McMaster University, Hamilton, Ontario
Besov spaces on complex balls: interpolating sequences and trees

The theory of interpolating sequences has its roots in the work of
Lennart Carleson on the corona problem. We characterize interpolating
sequences for Besov spaces on complex balls in dimension greater than
one, extending the recent one dimensional work of B. Boe, using a new
concept of holomorphic tree.
This is joint work with Nicola Arcozzi and Richard Rochberg.
 ALINA STANCU, Université de Montréal & Polytechnic University of NY
Some uniqueness results to a class of differential equations

Let g be a positive 2pperiodic function and let p be
some real number. Assuming solutions to the equation u^{1p} (u"+u) = g exist, we will discuss the use of curvature flows and convex
geometry in counting the number of solutions.
 PAUL TAYLOR, McMaster University, 1280 Main W, Hamilton, ON L8S 4K1
BochnerRiesz Means With Respect to a Rough Distance
Function

The generalized BochnerRiesz operator may be defined as as a Fourier
multiplier operator, where the multiplier is given by taking the
distance to a surface and raising this distance to a positive power.
The behaviour of the operator is described when the surface is taken
to be a cylinder, which results in a rough distance function.
 DIMITER VASSILEV, CRM/ISM and UQAM
Analytic continuation of the functions P^{l}

Let P be a function defined and continuous near the origin of
R^{N}. Consider the integral
J_{P} (l) = 
ó õ

B_{r}

P(x)^{l} dV, 
 (1) 
where dV is the Lebesgue measure on R^{N}. For a fixed
P the above integral converges absolutely when the real part of
l is positive. Furthermore, it defines a holomorphic function
of l in this region. A natural question is to find spaces of
functions for which J_{P} (l) has a holomorphic extension to the
left of the imaginary axis for any P in the considered space. One
can also ask if the extension is uniform with respect to "small"
perturbations of P.
We shall present an approach to the above questions for the space of
local solutions to an elliptic equation with Lipschitz coefficients.
 JIE XIAO, Memorial University of Newfoundland, St. John's, NL A1C 5S7
The Heat Equation I: CarlesonSobolev Estimates

We give Carleson type estimates for an operator valued solution of the
heat equation with initial data in the homogeneous Sobolev space.

