Maximal Operators in R^2
| dc.contributor.author | Choi, Keon | |
| dc.date.accessioned | 2007-08-13T18:58:49Z | |
| dc.date.available | 2007-08-13T18:58:49Z | |
| dc.date.issued | 2007-08-13T18:58:49Z | |
| dc.date.submitted | 2007 | |
| dc.description.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. | en |
| dc.format.extent | 512287 bytes | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | http://hdl.handle.net/10012/3159 | |
| dc.language.iso | en | en |
| dc.pending | false | en |
| dc.publisher | University of Waterloo | en |
| dc.subject.program | Pure Mathematics | en |
| dc.title | Maximal Operators in R^2 | en |
| dc.type | Master Thesis | en |
| uws-etd.degree | Master of Mathematics | en |
| uws-etd.degree.department | Pure Mathematics | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |