Proceedings of the American Mathematical Society

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



$ L\sp p$ norms of the Borel transform and the decomposition of measures

Author: B. Simon
Journal: Proc. Amer. Math. Soc. 123 (1995), 3749-3755
MSC: Primary 44A15; Secondary 28A10
MathSciNet review: 1277133
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Abstract: We relate the decomposition over [a, b] of a measure $ d\mu $ (on $ \mathbb{R}$) into absolutely continuous, pure point, and singular continuous pieces to the behavior of integrals $ \smallint\limits_a^b {{(\operatorname{Im} F(x + i\epsilon ))}^p}dx$ as $ \epsilon \downarrow 0$. Here F is the Borel transform of $ d\mu $, that is, $ F(z) = \smallint {(x - z)^{ - 1}}d\mu (x)$.

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  • [1] R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimension, rank one perturbations, and localizations (in preparation).
  • [2] Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/1966), 258–279. MR 0190801
  • [3] Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition, Dover Publications, Inc., New York, 1976. MR 0422992
  • [4] Abel Klein, Extended states in the Anderson model on the Bethe lattice, Adv. Math. 133 (1998), no. 1, 163–184. MR 1492789, 10.1006/aima.1997.1688
  • [5] Paul Koosis, Introduction to 𝐻_{𝑝} spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
  • [6] Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
  • [7] Barry Simon, Operators with singular continuous spectrum. I. General operators, Ann. of Math. (2) 141 (1995), no. 1, 131–145. MR 1314033, 10.2307/2118629
  • [8] Barry Simon, Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149. MR 1332038
  • [9] Tudor Zamfirescu, Most monotone functions are singular, Amer. Math. Monthly 88 (1981), no. 1, 47–49. MR 619420, 10.2307/2320713

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Article copyright: © Copyright 1995 American Mathematical Society