ESI Senior Research Fellow Program, spring term 2003
Recent trends in Fourier analysis
Course of advanced graduate lectures by
Professor Michael Lacey
(Georgia Institute of
Technology, Atlanta, USA)
Wednesday, 10:15 – 12:00, ESI lecture hall
14:00 – 15:00, Währinger Str. 17, floor 6 (corresponding seminar)
The course starts on March 5, 2003
A fundamental result of Lennart Carleson on the pointwise convergence
of Fourier series, from 1965, is related to the analysis of singular
integrals that have symmetries with respect to modulation. These
connections have only become apparent in the last few years, and the
subject has grown quite a bit since the connection came to light. This
course will investigate some parts ot this area.
1. Convolution and
discrete approximations - Introduce a notion of tiles, (closely
related to frames) and operators formed from sums of tiles.
Averaging, in appropriate ways, over tiles, generates convolution
operators. Special attention will be paid to the Hilbert transform.
Outline a connection between Littlewood---Paley theory, wavelets, BMO
spaces, and singular integrals.
2. Singular Integrals with Modulation - Carleson's Theorem on the
convergence of Fourier series. [Complete proof of the pointwise
convergence of square integrable Fourier series. Outline how this
proof should be modified to prove the convergence of Fourier series
of Lp functions, p>1.] Bilinear Hilbert
Transform.Complements and Extensions [including work of Lacey,
Thiele, Muscalu, Tao, Grafakos, Nahmod, Gilbert, Li, and Terwilliger,
3. Dyadic Models, T1 and Tb theorems - The dyadic models that
are used in recent works of Lacey and Thiele and coauthors, as well as
those of Nazarov, Treil, Volberg and coauthors, give a useful insight
into the T1 theorem of David and Journe. The Tb theorem has recently
been revisited, through the solution of the Kato Square Root Problem,
of Auscher, Hofmann, Lacey, McIntosh, and Tchmaitchin.
4. Weak-type Orthogonality and Littlewood-Paley inequalities -
We state a weak L2 orthogonality principle, which
arises from the proof of Carleson's theorem. The connection to
Littlewood-Paley inequalities is interesting, especially those
variants suggested by Rubio de Francia's inequality. Some of these
inequalities are proved in a situation governed by BMO in product
5.Commutator Estimates, Weak Factorization in H1 of
product domains - It is a classical fact that an H1
function can be factored into a product of functions in
H2. A recent result of Ferguson and Lacey, building on a
prior result of Ferguson and Sadosky, establishes a weak factorization
result for functions in H1 of product domain. This fact
entails significant difficulties not present in the classical case.
6. Recent thoughts on Hilbert transform on vector fields - A
well known problem concerns the boundedness of a Hilbert transform, or
Maximal function, computed in a unit line segment in the plane, whose
direction varies in a smooth fashion. A positive answer to this
question would depend upon a very delicate extension of Carleson's
theorem to a higher dimensional setting.
For further information please contact Prof. Hans Georg Feichtinger (Hans.Georg.Feichtinger@univie.ac.at)
ESI Senior Research Fellow Program coordinated by Prof. Joachim
Schwermer, Institut für Mathematik, Universität Wien,
Strudlhofgasse 4, A-1090 Wien (Joachim.Schwermer@univie.ac.at).