## Arkiv för Matematik

• Ark. Mat.
• Volume 38, Number 1 (2000), 183-199.

### On the T(1)-theorem for the Cauchy integral

Joan Verdera

#### Abstract

The main goal of this paper is to present an alternative, real variable proof of the T(1)-theorem for the Cauchy integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the T(1)-theorem. An example shows that the L-BMO estimate for the Cauchy integral does not follow from L2 boundedness when the underlying measure is not doubling.

#### Article information

Source
Ark. Mat. Volume 38, Number 1 (2000), 183-199.

Dates
First available in Project Euclid: 31 January 2017

http://projecteuclid.org/euclid.afm/1485898673

Digital Object Identifier
doi:10.1007/BF02384497

Zentralblatt MATH identifier
1039.42011

Rights

#### Citation

Verdera, Joan. On the T (1)-theorem for the Cauchy integral. Ark. Mat. 38 (2000), no. 1, 183--199. doi:10.1007/BF02384497. http://projecteuclid.org/euclid.afm/1485898673.

#### References

• David, G., Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. 17 (1984), 157–189.
• David, G., Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Math. 1465, Springer-Verlag, Berlin-Heidelberg, 1991.
• David, G., Unrectifiable 1-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), 369–479.
• David, G. and Mattila, P., Removable sets for Lipschitz harmonic functions in the plane, Preprint, 1997.
• Garnett, J., Analytic Capacity and Measure, Lecture Notes in Math. 297, Springer-Verlag, Berlin-Heidelberg, 1972.
• Journé, J.-L., Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math. 994, Springer-Verlag, Berlin-Heidelberg, 1983.
• Joyce, H. and Mörters, P., A set with finite curvature and projections of zero length, Preprint, 1997.
• Léger, J.-C., Courbure de Menger et rectifiabilité, Ph. D. Thesis, Université de Paris-Sud, 1997.
• Mattila, P., On the analytic capacity and curvature of some Cantor sets with non-σ-finite length, Publ. Mat. 40 (1996), 127–136.
• Mattila, P., Melnikov, M. S. and Verdera, J., The Cauchy integral, analytic capacity and uniform rectifiability, Ann. of Math. 144 (1996), 127–136.
• Melnikov, M. S., Analytic capacity: discrete approach and curvature of a measure, Mat. Sb 186:6 (1995), 57–76 (Russian). English transl.: Russian Acad. Sci. Sb. Math. 186 (1995), 827–846.
• Melnikov, M. S. and Verdera, J., A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices 1995, 325–331.
• Nazarov, F., Treil, S. and Volberg, A., Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1997, 703–726.
• Nazarov, F., Treil, S. and Volberg, A., Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998, 463–487.
• Stein, E. M., Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, N. J., 1993.
• Tolsa, X., Cotlar's inequality without the doubling condition and existence of principal values for the Cauchy integral of measures, J. Reine Angew. Math. 502 (1998), 199–235.
• Tolsa, X., Curvature of measures, Cauchy singular integral and analytic capacity, Thesis, Universitat Autònoma de Barcelona, 1998.
• Tolsa, X., L2-boundedness of the Cauchy integral operator for continuous measures, to appear in Duke Math. J.
• Verdera, J., A weak type inequality for Cauchy transforms of finite measures, Publ. Mat. 36 (1992), 1029–1034.
• Verdera, J., A new elementary proof of L2 estimates for the Cauchy Integral on Lipschitz graphs, Manuscript from a lecture given at the Conference on Geometrical and Algebraical Aspects in Several Complex Variables (Cetraro, 1994).