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 L^p functions, p>1.] Bilinear Hilbert Transform.Complements and Extensions [including work of Lacey, Thiele, Muscalu, Tao, Grafakos, Nahmod, Gilbert, Li, and Terwilliger, et al.]

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 domains.

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)