Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Existence and weak-type inequalities for Cauchy integrals of general measures on rectifiable curves and sets

Authors: Pertti Mattila and Mark S. Melnikov
Journal: Proc. Amer. Math. Soc. 120 (1994), 143-149
MSC: Primary 30E20; Secondary 42B20
MathSciNet review: 1160305
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $ \mu $ is a finite complex Borel measure and $ \Gamma $ a Lipschitz graph in the complex plane, then for $ \lambda > 0$

$\displaystyle \left\vert {\left\{ {z \in \Gamma :\mathop {\sup }\limits_{\varep... ...\right\vert \leqslant c(\Gamma ){\lambda ^{ - 1}}\vert\vert\mu \vert{\vert _1}.$

It follows that for any finite Borel measure $ \mu $ and any rectifiable curve $ \Gamma $ the finite principal value

$\displaystyle \mathop {\lim }\limits_{\varepsilon \downarrow 0} \int_{\vert\zeta - z\vert \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}d\mu \zeta } $

exists for almost all (with respect to length) $ z \in \Gamma $.

References [Enhancements On Off] (What's this?)

  • [C] A.-P. Calderón, Commutators, singular integrals on Lipschitz curves and applications, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 85–96. MR 562599
  • [CM] Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1104656
  • [D1] Guy David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 157–189 (French). MR 744071
  • [D2] -, Wavelets, Calderón-Zygmund operators, and singular integrals on curves and surfaces, Lecture Notes in Math., vol. 1465, Springer-Verlag, New York, 1991.
  • [F] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
  • [GT] T. W. Gamelin, Uniform algebras, Chelsea, New York, 1984.
  • [GJ] John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006
  • [GM] M. de Guzman, Differentiation of integrals in $ {\mathbb{R}^n}$, Lecture Notes in Math., vol. 481, Springer-Verlag, New York, 1975.
  • [K] Dmitry Khavinson, F. and M. Riesz theorem, analytic balayage, and problems in rational approximation, Constr. Approx. 4 (1988), no. 4, 341–356. MR 956172,
  • [M] Pertti Mattila, Singular integrals and rectifiability, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), 2002, pp. 199–208. MR 1964821,
  • [MT] Takafumi Murai, A real variable method for the Cauchy transform, and analytic capacity, Lecture Notes in Mathematics, vol. 1307, Springer-Verlag, Berlin, 1988. MR 944308
  • [V] Joan Verdera, A weak type inequality for Cauchy transforms of finite measures, Publ. Mat. 36 (1992), no. 2B, 1029–1034 (1993). MR 1210034,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30E20, 42B20

Retrieve articles in all journals with MSC: 30E20, 42B20

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society