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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Estimates for some Kakeya-type maximal operators


Author: Jose Barrionuevo
Journal: Trans. Amer. Math. Soc. 335 (1993), 667-682
MSC: Primary 42B25; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9947-1993-1150012-9
MathSciNet review: 1150012
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Abstract: We use an abstract version of a theorem of Kolmogorov-Seliverstov-Paley to obtain sharp ${L^2}$ estimates for maximal operators of the form: \[ {\mathcal {M}_\mathcal {B}}f(x) = \sup \limits _{x \in S \in \mathcal {B}} \frac {1}{{|S|}}\int _S {|f(x - y)|dy} \] . We consider the cases where $\mathcal {B}$ is the class of all rectangles in ${{\mathbf {R}}^n}$ congruent to some dilate of ${[0,1]^{n - 1}} \times [0,{N^{ - 1}}]$; the class congruent to dilates of ${[0,{N^{ - 1}}]^{n - 1}} \times [0,1]$ ; and, in ${{\mathbf {R}}^2}$ , the class of all rectangles with longest side parallel to a particular countable set of directions that include the lacunary and the uniformly distributed cases.


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Article copyright: © Copyright 1993 American Mathematical Society