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)



$ 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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.

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

  • [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] T. Kato, Wave operators and similarity for some non-self-adjoint operators, Math. Ann. 162 (1966), 258-279. MR 0190801 (32:8211)
  • [3] Y. Katznelson, An introduction to harmonic analysis, Dover, New York, 1976. MR 0422992 (54:10976)
  • [4] A. Klein, Extended states in the Anderson model on Bethe lattice, preprint. MR 1492789 (98k:82096)
  • [5] P. Koosis, Introduction to $ {H_p}$ spaces, London Math. Soc. Lecture Note Ser., vol. 40, Cambridge Univ. Press, New York, 1980. MR 565451 (81c:30062)
  • [6] M. Reed and B. Simon, Methods of modern mathematical physics, IV. Analysis of operators, Academic Press, New York, 1978. MR 0493421 (58:12429c)
  • [7] B. Simon, Operators with singular continuous spectrum, I. General operators, Ann. of Math. (2) 141 (1995), 131-145. MR 1314033 (96a:47038)
  • [8] -, Spectral analysis of rank one perturbations and applications, Proc. Mathematical Quantum Theory II: Schrödinger Operators (J. Feldman, R. Froese, and L. M. Rosen, eds.), Amer. Math. Soc., Providence, RI (to appear). MR 1332038 (97c:47008)
  • [9] T. Zamfirescu, Most monotone functions are singular, Amer. Math. Monthly 88 (1981), 47-49. MR 619420 (83a:26012)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 44A15, 28A10

Retrieve articles in all journals with MSC: 44A15, 28A10

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society