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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The limiting case of the Marcinkiewicz integral: growth for convex sets


Authors: N. Kruglyak and E. A. Kuznetsov
Journal: Proc. Amer. Math. Soc. 135 (2007), 3283-3293
MSC (2000): Primary 42B20, 42B25
Published electronically: June 20, 2007
MathSciNet review: 2322760
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Abstract: The Marcinkiewicz integral

$\displaystyle I_{\lambda }\left( x\right) =\underset{\Omega }{\int } \frac{\lef... ...} {\left\vert x-y\right\vert ^{n+\lambda }}dy\text{, where }\lambda >0\text{,} $

plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case $ \lambda \rightarrow 0$. Throughout, we assume that $ \Omega $ is convex; it is interesting that this condition cannot be dropped.


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Additional Information

N. Kruglyak
Affiliation: Department of Mathematics, LuleåUniversity of Technology, 971 87 Luleå, Sweden
Email: natan@ltu.se

E. A. Kuznetsov
Affiliation: Department of Mathematics, LuleåUniversity of Technology, 971 87 Luleå, Sweden
Email: evgeny@sm.luth.se

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08856-9
PII: S 0002-9939(07)08856-9
Keywords: Riesz potential, Marcinkiewicz integral, Poisson's equation
Received by editor(s): May 18, 2006
Received by editor(s) in revised form: July 13, 2006
Published electronically: June 20, 2007
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.