Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Weak $L^{1}$ norms of random sums

Author(s): Paul Alton Hagelstein
Journal: Proc. Amer. Math. Soc. 133 (2005), 2327-2334.
MSC (2000): Primary 42B35
Posted: March 4, 2005
MathSciNet review: 2138875
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Abstract: Let $\left\{g_{j}\right\}$ denote a sequence of measurable functions on $\mathbf{R}^{n}$ , and let $\left\Vert\cdot\right\Vert _{WL^{1}}$ denote the weak $L^{1}$ norm. It is shown that

$\begin{displaymath}\left\Vert\mathbb{E}\left(\left\vert\sum_{j=1}^{N} \epsilon_{... ... \lesssim \sum_{j=1}^{N} \left\Vert g_{j}\right\Vert _{WL^{1}},\end{displaymath}$

where $\left\{\epsilon_{j}\right\}$ is a sequence of independent random variables taking on values

and

with equal probability. Moreover, it is shown that

$\begin{displaymath}\left\Vert \mathbb{E}\left(\left\vert\sum_{j=1}^{N} \epsilon_... ...rt\sum_{j=1}^{N} \epsilon_{j}g_{j}\right\Vert _{WL^{1}}\right).\end{displaymath}$

The paper concludes by providing an example indicating that, if $\left\Vert g_{1}\right\Vert _{WL^{1}}$ $= \cdots = \left\Vert g_{N}\right\Vert _{WL^{1}} = 1$ , then the estimate

$\begin{displaymath}\mathbb{E}\left(\left\Vert\sum_{j=1}^{N}\epsilon_{j}g_{j}\right\Vert _{WL^{1}}\right) \lesssim N \log N\end{displaymath}$

is the best possible.

References:

1.: E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54. MR 0241685 (39:3024)
2.: A. Zygmund, Trigonometric Series, Cambridge University Press, 1959.MR 0107776 (21:6498)

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Additional Information:

Paul Alton Hagelstein
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798
Email: paul_hagelstein@baylor.edu

DOI: 10.1090/S0002-9939-05-07966-9
PII: S 0002-9939(05)07966-9
Received by editor(s): November 21, 2003
Posted: March 4, 2005
Additional Notes: The author's research was partially supported by the Baylor University Summer Sabbatical Program.
Communicated by: Andreas Seeger
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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