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PROPERTIES OF THE MAXIMAL OPERATORS ASSOCIATED WITH BASES OF RECTANGLES IN $\mathbb{R}^3$

Published online by Cambridge University Press:  28 July 2008

Alexander Stokolos
Affiliation:
Department of Mathematics, DePaul University, Chicago, IL 60614, USA ([email protected])
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Abstract

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This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions $(t,1/t,s)$ within a framework of more general Soria bases. The Jessen–Marcinkiewicz–Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps $L\log^2L$ into $L^{1,\infty}$. We give a simple geometric condition that guarantees that the $L\log^2L$ class cannot be enlarged. The proof develops the author's methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fefferman and Pipher.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2008