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## The Szegï¿½ Kernel for Non-Pseudoconvex Domains in â<sup>2</sup>

<p>There are many operators associated with a domain Î© â â<sup>n</sup> with smooth boundary âÎ©. There are two closely related projections that are of particular interest. The <i>Bergman projection</i> <b><i>B</b></i> is the orthogonal projection of L<sup>2</sup>(Î©) onto the closed subspace L<sup>2</sup>(Î©)â©O(Î©), where O(Î©)is the space of all holomorphic functions on â¦. The <i>Szegï¿½ projection</i> <b><i>S</b></i> is the orthogonal projection of L<sup>2</sup>(ââ¦) onto the space H<sup>2</sup>(Î©) of boundary values of elements of O(Î©). These projection operators have integral representations</p>

<p><b><i>B</b></i>[f](z) = <big>â«</big><sub>â¦,</sub>f(w)<b><i>B</b></i>(z,w)dw, <b><i>S</b></i>[f](z) = <big>â«</big><sub>ââ¦,</sub>f(w)<b><i>S</b></i>(z,w)do(w).</p>

<p>The distributions <b><i>B</b></i> and <b><i>S</b></i> are known respectively as the Bergman and Szegï¿½ kernels. In an attempt to prove that <b><i>B</b></i> and <b><i>S</b></i> are bounded operators on L<sup>p</sup>, 1 < p < â, many authors have obtained size estimates for the kernels B and S for <i>pseudoconvex</i> domains in â<sup>n</sup>.</p>

<p>In this thesis, we restrict our attention to the Szegï¿½ kernel for a large class of domains of the form 1 Such a domain fails to be pseudoconvex precisely when b is not convex on all of R. In an influential paper, Nagel, Rosay, Stein, and Wainger obtain size estimates for both kernels and sharp mapping properties for their respective operators in the convex setting. Consequently, if b is a convex polynomial, the Szegï¿½ kernel S is absolutely convergent off the diagonal only. Carracino proves that the Szegï¿½ kernel has singularities on <i>and off</i> the diagonal for a specific non-smooth, <i>{non-convex</i> piecewise defined quadratic b. Her results are novel since very little is known for the Szegï¿½ kernel for non-pseudoconvex domains 2. I take b to be an arbitrary even-degree polynomial with positive leading coefficient and identify the set in 3 on which the Szegï¿½ kernel is absolutely convergent. For a polynomial b, we will see that the Szegï¿½ kernel is smooth off the diagonal if and only if b is convex. These results provide an incremental step toward proving the projection S is bounded on 4, for a large class of non-pseudoconvex domains â¦.</p>

Identifer | oai:union.ndltd.org:MONTANA/oai:etd.lib.umt.edu:etd-08032011-102211 |

Date | 05 August 2011 |

Creators | Gilliam, Michael |

Contributors | Jennifer Halfpap, Eric Chesebro, Michael Schneider, Karel Stroethoff, Thomas Tonev |

Publisher | The University of Montana |

Source Sets | University of Montana Missoula |

Language | English |

Detected Language | English |

Type | text |

Format | application/pdf |

Source | http://etd.lib.umt.edu/theses/available/etd-08032011-102211/ |

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