Maximal Operators in R^2
Loading...
Date
2007-08-13T18:58:49Z
Authors
Choi, Keon
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
A maximal operator over the bases $\mathcal{B}$ is defined as
\[Mf(x) = \sup_{x \in B \in \mathcal{B}} \frac{1}{|B|}\int_B |f(y)|dy. \]
The boundedness of this operator can be used in a number of applications including the Lebesgue differentiation theorem. If the bases are balls or rectangles parallel to the coordinate axes, the associated maximal operator is bounded from $L^p$ to $L^p$ for all $p>1$. On the other hand, Besicovitch showed that it is not bounded if the bases consists of arbitrary rectangles. In $\mathbb{R}^2$ we associate a subset $\Omega$ of the unit circle to the bases of rectangles in direction $\theta \in \Omega$. We examine the boundedness of the associated maximal operator $M_{\Omega}$ when $\Omega$ is lacunary, a finite sum of lacunary sets, or finite sets using the Fourier transform and geometric methods. The results are due to Nagel, Stein, Wainger, Alfonseca, Soria, Vargas, Karagulyan and Lacey.