Abstracts

Harmonic Analysis and Related Topics

Retractions and projections for Chebyshev subsets of function and sequence spaces
by
Sergey Ajiev
University of New South Wales

Along with Lebesgue and sequence spaces with mixed norms, anisotropic Besov, Lebesgue, Lizorkin-Triebel and Sobolev spaces of differentiable functions defined on a domain and endowed with various norms are considered. We estimate the constants and determine the exponents for the local Hölder regularity of the Chebyshev centres, metric projections and some retractions for the closed convex subsets of these spaces. Attention is paid to the sharpness of some results.

Date received: October 30, 2008

Sobolev Spaces and Fractional Smoothness Spaces on Irregular Domains
by
Oleg V. Besov
Steklov Mathematical Institute

Function spaces on irregular domains of certain type of multidimensional Euclidean spaces are studied. Such domains may have, for instance, the shape of the external peak with a power-like degeneracy in the neighborhood of some boundary point. The embedding theorems for these spaces are established.

Date received: October 30, 2008

Characterization of function spaces via non-smooth kernels
by
Qui Bui
University of Canterbury
Coauthors: Tim Candy

The characterization of function spaces via a kernel in the Schwartz class S was established by Bui, Paluszinski and Taibleson in the mid-1990's. In this talk, using the concept of a bounded distribution introduced by E. Stein, I will present an extension of this characterization to the case where the kernel is not in S. This is joint work with Tim Candy.

Date received: October 30, 2008

Hardy's uncertainty principle
by
Michael Cowling
University of Birmingham (UK)

Hardy showed that if |f(x)| ≤ C exp(-a|x|²) and |[^f](x)| ≤ C exp(-a|x|²), where a, a > 0, then f=0 if aa is big enough, and if aa is the critical value, then f is a gaussian.

This has recently been extended in various ways, for instance to deal with operators (by Cowling, Demange, and Sundari; to appear). This talk surveys the inequality and some new extensions.

Date received: October 31, 2008

Contractions of Lie Groups and the Cowling-Haagerup Theorem
by
Anthony Dooley
University of New South Wales

Inönü and Wigner introduced the notion of a contraction, or continuous deformation, of one Lie group into another (generally non-isomorphic) Lie group. I discuss here the contraction of compact Lie groups arising as the K in the Iwasawa decomposition of rank one semi-simple group G, into the associated generalised Heisenberg group. It turns out that the techniques developed can be used to describe the relationship between the K-picture and the N-picture of the representation theory of G, and this has consequences for the Cowling-Haagerup constants.

Date received: November 3, 2008

Multilinear operators with non-smooth kernels and commutators of singular integrals
by
Xuan Thinh Duong
Macquarie university
Coauthors: Loukas Grafakos and Lixin Yan

We obtain endpoint estimates for multilinear singular integrals operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calderón-Zygmund kernels. As a consequence, we deduce endpoint L¹ ×... ×L¹ to weak L^1/m estimates for the mth order commutator of Calderón. Our results reproduce known estimates for m = 1, 2 but are new for m ≥ 3. We also explore connections between the 2nd order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.

Date received: October 30, 2008

Invariant subspaces of submarkovian semigroups
by
Tom ter Elst
University of Auckland
Coauthors: Derek Robinson

If S is a submarkovian semigroup acting on a Hilbert space L₂(X) and W is a measurable subset of X then we characterize the invariance of L₂(W) by capacity conditions on the boundary of W.

Date received: October 27, 2008

Frequency-scale frames and the solution of the Mexican hat problem
by
Richard S. Laugesen
University of Illinois
Coauthors: H.-Q. Bui

Yves Meyer revealed the depth of our ignorance of non-orthogonal wavelets when he remarked, in his monograph on wavelets and operators, that we do not even know whether the Mexican hat wavelet system is complete in L^p for 1 < p < ∞. Completeness has been known only for p=2, by the sufficient frame condition of Daubechies. I have proved completeness for p > 2, in joint work with H.-Q. Bui. The talk will describe this wavelet spanning problem and the tools we develop to solve it, which include approximately dual frames in the Fourier domain.

Date received: October 26, 2008

Duality of Hardy space on product spaces of homogeneous type
by
Ji Li
Mathematics Department, Macquarie University
Coauthors: Yongsheng Han, Guozhen Lu

In this paper, we introduce the Carleson measure space CMOp on product spaces of homogeneous type in the sense of Coifman and Weiss, and prove that it is the dual space of the product Hardy space Hp of two homogeneous spaces. Our results thus extend the duality theory of Chang and R. Fefferman on the bi-disc, which was established using bi-Hilbert transform. Our method is to use discrete Littlewood-Paley-Stein theory in product spaces.

Date received: September 24, 2008

An inequality for bi-orthogonal pairs
by
Christopher Meaney
Macquarie University

We use ideas from Salem's proof of the Rademacher-Menshov Theorem, combined with a result of Menshov, to give a logarithmic lower bounds on sums of vectors in bi-orthogonal pairs. This is applied to estimates on Lebesgue functions for orthogonal expansions.

Date received: October 23, 2008

Using solid angles to get approximate measures of volume for polytopes
by
Sinai Robins
Nanyang Technological University
Coauthors: David Desario

Using lattices in R^d, we compute local contributions (solid angles) at each lattice point and find an interesting new Fourier series for their global sum over any real polytope, extending a few known results of I.G. Macdonald and A. Barvinok.

The global sums that we encounter form a different sort of discrete measure for the volume for any given real polytope. When p=2, we retrieve the classical solid angle polynomial of Macdonald, which turns out to be an integral of a classical theta function over a polytope. For other p's we have new variations on the theme of discrete volumes. This variation on a theme also extend the notion of a spherically symmetric solid angle to an L^p-ball solid angle, computed locally at each integer point inside a convex polytope The methods involve Fourier analysis. I'll define inasmuch as it is possible much all of the notions we encounter.

Date received: October 21, 2008

Bochner-Riesz analysis on on asymptotically conic manifolds
by
Adam Sikora
Australian National University
Coauthors: Joint work with Colin Guillarmou and Andrew Hassell

Let (M, g) be a complete noncompact manifold with the Riemannian metric g which is asymptotically conic in the sense that M compactifies to a manifold with boundary M' in such a way that g becomes a scattering metric on M'. Let D be the positive Laplacian associated to g, and L = D+ V, where V is a potential function obeying certain conditions. We analyze the spectral measure dE_L(l) = [1/(2pi)] R(l+i0) - R(l- i0), where R(l) = (L - l²)^-1. We obtain L¹ → L^∞ estimates on derivatives (in l) of the spectral measure. Hence, under assumption that the manifolds M is nontrapping, we obtain restriction theorems, i.e. L^p → L^p' mapping properties of the spectral projections, which are as good as those currently known for flat Euclidean space. As an immediate application, we prove spectral multiplier and Bochner Riesz summability results for L, similar to those known for the standard Laplace operator.

Date received: November 2, 2008

Symmetric norms and spaces of operators
by
F. Sukochev
University of New South Wales
Coauthors: N. Kalton

In 1937, von Neumann showed that if ∥·∥_E is a symmetric norm on Rⁿ then one can define a norm on the space of n×n matrices by

∥A∥E = ∥(s1(A), ..., sn(A))∥E

where s₁(A), ..., s_n(A) are the singular values of A (i.e. the eigenvalues of (A^*A)^1/2 in decreasing order. Surprisingly, the infinite-dimensional analogue of this result, although well-known in special cases, has never been established in complete generality. Very recently, in a joint work with N. Kalton, we have shown that (E, ∥·∥_E) is a symmetric Banach sequence space then the corresponding space S_E of operators on a separable Hilbert space, defined by T ∈ S_E if and only if (s_n(T))_n=1^∞ ∈ E, is a Banach space under the norm ∥T∥_SE=∥(s_n(T))_n=1^∞∥_E thus providing complete infinite-dimensional extension of von Neumann's result.

Date received: October 24, 2008

Harmonic measure distribution functions for sequences of planar domains
by
Lesley A. Ward
University of South Australia
Coauthors: Marie A. Snipes

The harmonic measure distribution function h(r), for r ≥ 0, of a planar domain D specifies the harmonic measure of the part of the boundary of D that lies within distance r of a fixed basepoint in D. It thus relates the geometry of the domain to the behaviour of Brownian motion in the domain. We establish sufficient conditions under which these functions h_n for a sequence of domains D_n converge pointwise to the function h for a limiting domain D, at all points of continuity of h. We establish this convergence for a model example.

Date received: October 23, 2008

Comparison of solutions of the heat and Laplace equations
by
Neil Watson
University of Canterbury

Solutions of the heat equation have sometimes been used in harmonic analysis, in preference to harmonic functions. There are some advantages in doing this, as larger functions can be handled, and the kernel for the infinite strip is the same as that for the half-space. In this talk, I will present some comparisons between the solutions of the heat and Laplace equations, particularly for the case of the half-space.

Date received: November 2, 2008