dc.contributor.author Choi, Keon dc.date.accessioned 2007-08-13 18:58:49 (GMT) dc.date.available 2007-08-13 18:58:49 (GMT) dc.date.issued 2007-08-13T18:58:49Z dc.date.submitted 2007 dc.identifier.uri http://hdl.handle.net/10012/3159 dc.description.abstract A maximal operator over the bases $\mathcal{B}$ is defined as en $Mf(x) = \sup_{x \in B \in \mathcal{B}} \frac{1}{|B|}\int_B |f(y)|dy.$ 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. dc.format.extent 512287 bytes dc.format.mimetype application/pdf dc.language.iso en en dc.publisher University of Waterloo en dc.title Maximal Operators in R^2 en dc.type Master Thesis en dc.pending false en dc.subject.program Pure Mathematics en uws-etd.degree.department Pure Mathematics en uws-etd.degree Master of Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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