# SUMMER SCHOOL T(1) and T(b) theorems and applications.

T(1) and T(b) theorems provide criteria for singular integral operators to be bounded in L2. In essence, boundedness is tested on a subset of functions on L2, and boudnedness is inferred for all functions of L2. The test fucntions are suitable bump functions adapted to intervals (one dimension) or squares (several dimension), and this localization principle stems from the localization properties of the singular integral kernels. Some of the highlight applciations of such theorems developed in the summer school are on analytic capacity (Painleve problem), square root of accretive operators (Kato's problem) and weighted estimates for singular integrals. T(1) theorems have been around for 30 years and remain a very active research area.

1. A boundedness criterion for generalized Calderón-Zygmund operators.
by David, Guy; Journé, Jean-Lin
Ann. of Math. (2) 120 (1984), no. 2, 371-397.
This is the paper that inspired the name and subject of T(1) theorems. The main result and proof on pages 371-379 should be presented. The applications in the rest of the paper will be discussed in other presentations of the summer school and can be omitted in this presentation. Some details should be filled in from other sources, in particular the detail on page 371 that is referred to [5] p. 149. This and all other details can also be found in the textbook "Harmonic Analyis" by E. Stein.
[presenter: Jingxin Zhong]
2. Two elementary proofs of the L2 boundedness of Cauchy integrals on Lipschitz curves.
by Coifman, R. R.; Jones, Peter W.; Semmes, Stephen
J. Amer. Math. Soc. 2 (1989), no. 3, 553-564.
The beauty in the comparison of the two proofs should be preserved in the presentation. Both proofs should be presented in fair detail and compared, with a clear mapping which steps correspond to each other. The first proof uses complex analysis which is more unique to this presentation, while the use of adapted Haar functions in the second proof will reappear in other presentations. The remarks about more general T(b) theorems are needed in Christ's paper and should be briefly scetched as time allows.
[presenter: Marco Vitturi]
3. Rectifiable sets and the traveling salesman problem.
by Jones, Peter W.
Invent. Math. 102 (1990), no. 1, 1-15.
Th paper should be covered in fair detail. The material will be used in the papers by Mateu Tolsa Verdera and Tolsa.
[presenter: Marti Prats]
4. A T(b) theorem with remarks on analytic capacity and the Cauchy integral.
by Christ, Michael
Colloq. Math. 60/61 (1990), no. 2, 601-628.
The construction of general dyadic cubes is a widely cited part of this paper and should be presented carefully along with the main result. The reference to Corifman Jones Semmes could be coordinated with the presenter of that paper. The remarks on analytic capacity shoudl be coordinated with the presentation of the Mattila Melnikov verdera paper who need these remarks. will be discussed in other presentations and the remarks in this apper should be very briefly presented with eye on the other presentations. For further inspiration on the topic of dyadic cubes on homogeneous spaces the presenter may consult the paper by Auscher/Hytönen "Orthonormal bases of regular wavelets in spaces of homogeneous type" on the arxiv.
[presenter: Sebastian Stahlhut]
5. The Cauchy integral, analytic capacity, and uniform rectifiability.
by Mattila, Pertti; Melnikov, Mark S.; Verdera, Joan
Ann. of Math. (2) 144 (1996), no. 1, 127-136.
The theorem by David Semmes used in this paper could be briefly summarized, but likely time will not suffice to give an in-depth discussion of it. We may take it as black box, and compare with the paper by Jones. The result by Christ should be coordinated with the presentation of that paper. Also presented should be the main result of A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs
by Melnikov, Mark S.; Verdera, Joan
Internat. Math. Res. Notices 1995, no. 7, 325-331.
[presenter: Joris Roos]
6. Painleve's problem and the semiadditivity of analytic capacity.
by Tolsa, Xavier
Acta Math. 190 (2003), no. 1, 105-149.
This is a culmination of previous presentations on analytic capacity. The paper follows the strategy of the special case presented in The planar Cantor sets of zero analytic capacity and the local T(b)-theorem.
Mateu, Joan; Tolsa, Xavier; Verdera, Joan
J. Amer. Math. Soc. 16 (2003), no. 1, 19-28
It may be a good idea to present the general ideas in the special case in the first talk, and discuss the generalizations needed in Tolsa's paper in the second talk.n Possibly not all details can be done thoroghly of Tolsa's paper. By the time of the second talk, sufficient familiarity with the paper by Nazarov Treil and Volberg can be assumed; it is cited heavily in this paper.
[presenter: Daniel Girela]
7. A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash.
by Fabes, E. B.; Stroock, D. W.
Arch. Rational Mech. Anal. 96 (1986), no. 4, 327-338.
This paper proves Gaussian estimates that are used in some papers (not presented here though) on Kato's problem. Thsi is an improtant paper of separate interest and should be discussed thorouighly. For general preparation to the summer school presenter should be familiar with at least one other paper on Kato's problem and If (and only if) time allows, might add some remarks on more general cases taken from the book of Auscher Tchamitchian, though time will not be enough for anywhere extensive discussion of such topics.
[presenter: Sukjung (Sue) Hwang ]
8. Ondelettes et conjecture de Kato.
by P. Auscher and P. Tchamitchian
C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 2, 63-66.
The article refers to the following two papers, which are also part of the assigment in so far as needed to complete the proof of the main theorem in the above paper. Ondelettes et intégrale de Cauchy sur les courbes lipschitziennes.
by P. Tchamitchian
Ann. of Math. (2) 129 (1989), no. 3, 641-649.
(This paper has some overlap with the simultaneous paper by Coifman, Jones, Semmes) Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs.
by J.-L. Lyons
J. Math. Soc. Japan 14 1962 233-241.
The presentation will be in English, despite all papers being in French.
[presenter: Yi Huang]
9. The solution of the Kato square root problem for second order elliptic operators on Rn.
by Auscher, Pascal; Hofmann, Steve; Lacey, Michael; McIntosh, Alan; Tchamitchian, Ph.
Ann. of Math. (2) 156 (2002), no. 2, 633-654.
This is a sophisticated version of a T(b) theorem, compare with the previous presentation on the paper by Auscher and Tchamitchian.
[presenter: Ana Grau de la Herran]
10. The Tb-theorem on non-homogeneous spaces. Part I.
by Nazarov, F.; Treil, S.; Volberg, A.
Acta Math. 190 (2003), no. 2, 151-239.
Pages 152-205 of this paper. Presenter should be aware of the material in the above paper by Christ, which does the homogeneous space case, and ofcourse coordinate with presenter of part II. The paper is used in the paper by Tolsa.
[presenter: Polona Durcik]
11. The Tb-theorem on non-homogeneous spaces. Part II.
by Nazarov, F.; Treil, S.; Volberg, A.
Acta Math. 190 (2003), no. 2, 151-239.
Present, pages 206-239 of this paper, the proof of the T(b) theorem. See remarks on Part I.
[presenter: Shaoming Guo]
12. On the Two Weight Hilbert Transform Inequality
by Michael T Lacey
arXiv:1301.4663
Section 4 is the main novelty, and should be focused on. The reduction in Section 3 refers to several related papers in the literature, the most important aspects should but likely not all of it can be summarized in the presentation.
[presenter: Vjekoslav Kovac]
13. Sharp weighted estimates for dyadic shifts and the \$A_2\$ conjecture
by Tuomas Hytönen, Carlos Pérez, Sergei Treil, Alexander Volberg
arXiv:1010.0755
[presenter: Timo Hänninen]
14. Boundedness of the twisted paraproduct
by Vjekoslav Kovac
arXiv:1011.6140
[presenter: Michal Warchalski]
15. Lp theory for outer measure etc.
by Y. Do, C. Thiele
preprint
[presenter: Mariusz Mirek]
16. The local Tb theorem with rough test functions.
by Tuomas Hytönen, Fedor Nazarov
http://arxiv.org/abs/1206.0907
[presenter: Paco Villarroya]