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)

 
 

 

A generalized Kolmogorov inequality for the Hilbert transform


Author: Mark A. Pinsky
Journal: Proc. Amer. Math. Soc. 130 (2002), 753-758
MSC (2000): Primary 42A50; Secondary 44A15
DOI: https://doi.org/10.1090/S0002-9939-01-06122-6
Published electronically: August 28, 2001
MathSciNet review: 1866030
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

If $f\in L^1(\mathbf{R}^1;(1+\vert x\vert)^{-1}dx)$ we can define the Hilbert transform $Hf$ almost everywhere (Lebesgue) and obtain an estimate for $\mu\{x:\,\vert Hf(x)\vert\ge \alpha\}$ where $\mu$ is a suitable finite measure. The classical Kolmogorov inequality for the Lebesgue measure of $\{x:\,\vert Hf(x)\vert\ge\alpha\}$ is obtained by a scaling argument.


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

  • [C] C. Choi, A weak-type inequality for differentially subordinate harmonic functions, Transactions of the American Mathematical Society, 350(1998), 2687-2696. MR 99e:31006
  • [D] B. Davis, On the distribution of conjugate functions of non-negative measures, Duke Mathematical Journal, 40(1973), 695-700. MR 48:2649
  • [G] J. Garnett, Bounded Analytic Functions, Academic Press, 1981. MR 83g:30037
  • [Kz] Y. Katznelson, Introduction to Harmonic Analysis, Dover reprint, 1976. MR 54:10976

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42A50, 44A15

Retrieve articles in all journals with MSC (2000): 42A50, 44A15


Additional Information

Mark A. Pinsky
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730
Email: pinsky@math.nwu.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06122-6
Received by editor(s): March 14, 2000
Received by editor(s) in revised form: September 11, 2000
Published electronically: August 28, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society