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Harmonic Analysis / Analyse harmonique
Org: Izabella Laba (UBC) and/et Alex Iosevich (Missouri-Columbia)

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MICHAEL CHRIST, University of California, Berkeley, California, USA
On multilinear oscillatory integrals

L^p norm inequalities are established for multilinear integral operators of Calderón-Zygmund type which incorporate oscillatory factors e^iP, where P is a real-valued polynomial. Our main results concern nonsingular multilinear operators L_l(f₁,f₂,...,f_n) = ò_R^m e^ilP(x) Õ_j=1ⁿ f_j(p_j(x))h(x) dx, where l Î R, P is a measurable real-valued function, each f_j Î L^¥, and h Î C¹₀ is compactly supported. Each p_j denotes the orthogonal projection from R^m to a linear subspace of R^m of arbitrary dimension k £ m-1. 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(Rⁿ)

We study the spaces Q_a(Rⁿ), which are subpaces of BMO, and their dyadic counterparts. We prove a quasi-orthogonal decomposition for functions in Q_a(Rⁿ), 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 Calderón-Zygmund SIO's on the Heisenberg group

We consider the class of singular integral operators on the Heisenberg group with radial convolution kernels satisfying standard Calderón-Zygmund-type conditions. In particular, this class includes the Cauchy-Szegö projection. Since their kernels are radial, the operators in this class can also be described as spectral multipliers. They form a sub-class of the Marcinkiewicz-type spectral multipliers, whose convolution kernels were characterized by Müller, Ricci, and Stein in 1995. I establish a condition on the multipliers which characterizes this sub-class.

SINAN GUNTURK, New York University, Courant Institute, New York, New York  10012, USA
Accuracy of one-bit quantization and the fair-duel problem: some extremal problems on {-1,+1} sequences

One-bit quantization refers to a class of algorithms widely used in analog-to-digital conversion to approximate bandlimited functions by local averages of {-1,+1} sequences on dense grids. The fair-duel problem (which the speaker heard from S. Konyagin) asks for a universal ordering of shootings that makes a duel between two equal and bad-shooter 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 Rⁿ, or the Laplace-Beltrami operator on a complete non-compact 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 L² estimate, which for the Laplace-Beltrami operator is immediate from self-adjointness, 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 Calderon-Zygmund 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

Hvf(x):=p.v.  ó õ 1 -1  f æ è x-yv(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 L²(R²) 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 non-smooth Riemannian manifolds

We study the well-posedness of the Dirichlet and Neumann problems for the Laplace-Beltrami operator in a Lipschitz sub-domain 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 Dini-type 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 |an| £ 1  || å n Î Ew  an e2pi n x||p £ Ce Ne|| å n Î Ew  e2pi 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 non-linear PDE

We will present a generalization of the well known inequality of Kato and Ponce in non-linear 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 h-fold 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 h-fold 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 R³

In this work joint with Andreas Seeger, we consider the L^p regularity of an averaging operator over a curve in R³ 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 Wisconsin-Madison
Maximal functions associated with Mikhlin-Hoermander multipliers

We consider Fourier multipliers satisfying a condition of Mikhlin-Hörmander type,

||fm(2k·)||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 Dirichlet-to-Neumann 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 Dirichlet-to-Neumann (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.