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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
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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, Proc. Internat. Congr. Math. (Helsinki, 1978), Acad. Sci. Fenn., Helsinki, 1980, pp. 85-96. MR 562599 (82f:42016)
  • [CM] M. Christ, Lectures on singular integral operators, CBMS Regional Conf. Ser. in Math., vol. 77, Amer. Math. Soc., Providence, RI, 1990. MR 1104656 (92f:42021)
  • [D1] G. David, Opérateurs integraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), 157-189. MR 744071 (85k:42026)
  • [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, Geometry of fractal sets, Cambridge Univ. Press, Cambridge, 1985. MR 867284 (88d:28001)
  • [GT] T. W. Gamelin, Uniform algebras, Chelsea, New York, 1984.
  • [GJ] J. B. Garnett, Analytic capacity and measure, Lecture Notes in Math., vol. 297, Springer-Verlag, New York, 1972. MR 0454006 (56:12257)
  • [GM] M. de Guzman, Differentiation of integrals in $ {\mathbb{R}^n}$, Lecture Notes in Math., vol. 481, Springer-Verlag, New York, 1975.
  • [K] D. Khavinson, F. and M. Riesz theorem, analytic balayage, and problems in rational approximation, Constr. Approx. 4 (1988), 341-350. MR 956172 (90e:30039)
  • [M] P. Mattila, Cauchy singular integrals and rectifiability of measures in the plane, Adv. in Math. (to appear). MR 1964821 (2004b:42032)
  • [MT] T. Murai, A real variable method for the Cauchy transform and analytic capacity, Lecture Notes in Math., vol. 1307, Springer-Verlag, New York, 1988. MR 944308 (89k:30022)
  • [V] J. Verdera, A weak type inequality for Cauchy transforms of measures, Publ. Mat. 36 (1992), 1029-1034. MR 1210034 (94k:30099)

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