Geodesic
The Poisson Integral of a Singular Measure
Canadian journal of mathematics, Tome 35 (1983) no. 4, pp. 735-749

Voir la notice de l'article provenant de la source Cambridge University Press

Let σ be a finite positive singular Borel measure defined on Euclidean space R N . For w ∈ R N and y > 0, its Poisson integral is defined by the formula where CN is chosen so that Since σ is singular, almost everywhere with respect to Lebesgue measure on R N . On the other hand, almost everywhere dσ. It follows that for all sufficiently small y, is a non-empty open subset of R N . If σ has compact support then |E y | → 0 as y → 0, where |E y | denotes the Lebesgue measure of E y . In this paper we give a lower bound on the rate at which |E y | may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |E y | may approach 0. 
Ahern, Patrick. The Poisson Integral of a Singular Measure. Canadian journal of mathematics, Tome 35 (1983) no. 4, pp. 735-749. doi: 10.4153/CJM-1983-042-0
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[1] 1. Ahern, P., The mean modulus and the derivative of an inner junction, Indiana University Mathematics Journal 28 (1979), 311–347. Google Scholar

[2] 2. Ahern, P. and Clark, D., In inner functions with BP derivative, Michigan Mathematics Journal 23 (1976), 107–118. Google Scholar

[3] 3. Besicovitch, A., A general form of the covering principle and relative differentiation of additive functions, Proc. Cambridge Philos. Soc. 41 (1945), 103–110. Google Scholar

[4] 4. Gluchoff, A, The mean modulus of a Blaschke product, Thesis, University of Wisconsin (1981). Google Scholar

[5] 5. Hardy, G., Littlewood, J. and Polya, G., Inequalities (Cambridge, 1934). Google Scholar

[6] 6. Jevtić, M., Sur la derivée de la fonction atomique, C. R. Acad. Se. Paris 292 (1981), 201–203. Google Scholar

[7] 7. Kahane, J. -P. and Salem, R., Ensembles parfait et séries trigonometrique (Hermann, 1963). Google Scholar

[8] 8. Rudin, W., Real and complex analysis, 2nd Ed. (McGraw-Hill, New York, 1974). Google Scholar

[9] 9. Walter, W., A counterexample in connection with Egorov's theorem, American Math. Monthly 84 (1977), 118–119. Google Scholar

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