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Estimates for some Kakeya-type maximal operators

Author: Jose Barrionuevo
Journal: Trans. Amer. Math. Soc. 335 (1993), 667-682
MSC: Primary 42B25; Secondary 47B38
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:

$\displaystyle {\mathcal{M}_\mathcal{B}}f(x) = \mathop {\sup }\limits_{x \in S \in \mathcal{B}} \frac{1}{{\vert S\vert}}\int_S {\vert f(x - y)\vert 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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