Instructor: Steve Hofmann

Description: The course will concentrate on so-called T1 and Tb theorems and their applications. Such theorems originally arose in connection with an attempt to better understand the Cauchy integral operator on Lipschitz curves. The circle of ideas related to these theorems and their proofs has subsequently found application to various problems in harmonic analysis and PDE, including the solution of the square root problem of Kato, and the solution of Vitushkin's conjecture on analytic capacity.

Here is a rough outline of topics that we plan to cover.

• T1 Theorem of David and Journe
• L² boundedness of Cauchy integral operator on Lipschitz curves (Coifman-McIntosh-Meyer Theorem) following ideas of P. Jones and M. Christ.
• Simple Tb Theorem of S. Semmes
• Full Tb Theorem (David-Journe-Semmes)
• Cauchy integral via Tb Theorem
• Solution of the Kato square root problem
• M. Christ's Tb Theorem and analytic capacity

and, time permitting, we'll discuss also some elements of the following:

• uniform rectifiability, Mattila-Melnikov-Verdera Theorem and the solution of Vitushkin's conjecture (by G. David in the case of finite 1-D Hausdorff measure, and by Tolsa in general).

Prerequisite: The rudiments of classical harmonic analysis in Rⁿ: basic theory of the Fourier transform, Hardy-Littlewood maximal function, approximate identities, Littlewood-Paley Theory, classical Calderon-Zygmund theory, BMO and Carleson measures. Enrollment in Harmonic Analysis I: Math 8630 (415) during fall semester will suffice.