Type: Article
Publication Date: 1993-02-01
Citations: 6
DOI: https://doi.org/10.2307/2154399
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.