Maximal Operators in R^2

[摘要] 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 $heta in Omega$. 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.

[发布机构] University of Waterloo

[关键词] Pure Mathematics