This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class $C^2$ in $\\mathbb{C}^N$, the Henkin-Ram\\'irez Kernel Function belongs to the Smirnov class, $E^q(D)$, for every $q\\in(0,N)$.
The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\\subset\\mathbb{C}$ of class $C^2$ if we fix $\\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function
\\mathcal{K}(\\zeta,z)=\\frac{1}{2\\pi i}\\frac{1}{\\zeta-z}
as a function of $z$, then the Cauchy Kernel Function belongs to the Smirnov class $E^q(D)$ for every $q\\in(0,1)$.
","abstract_html":"<p>This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class C2 in \\mathbb{C}N, the Henkin-Ram\\'irez Kernel Function belongs to the Smirnov class, Eq(D), for every $q\\in(0,N)$.</p> <p>The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\\subset\\mathbb{C}$ of class C2 if we fix $\\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function</p> <p>\\mathcal{K}(\\zeta,z)=\\frac{1}{2\\pi i}\\frac{1}{\\zeta-z}</p> <p>as a function of $z$, then the Cauchy Kernel Function belongs to the Smirnov class Eq(D) for every $q\\in(0,1)$.</p>","abstract_has_math":true,"creators":["Espinosa Lucio, Belen"],"institution":null,"degree_name":"Doctor of Philosophy in Mathematics (PhD)","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Harrington, Phillip S.","Luecking, Daniel H."],"advisors":["Lanzani, Loredana"],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-01T07:00:00Z","date_published":"2013-05-01T07:00:00Z","updated_at":"2026-07-24T00:59:32Z","subjects":["Pure sciences","Applied sciences","Cauchy kernel","Complex variables","Hardy spaces","Smirnov space","Numerical Analysis and Computation"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.uark.edu/etd/766","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Harrington, Phillip S.","Luecking, Daniel H."]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Lanzani, Loredana"]},{"key":"dc:creator","label":"Author","values":["Espinosa Lucio, Belen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-09-29T07:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy in Mathematics (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Applied sciences","Cauchy kernel","Complex variables","Hardy spaces","Smirnov space","Numerical Analysis and Computation"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.uark.edu/etd/766"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class $C^2$ in $\\mathbb{C}^N$, the Henkin-Ram\\'irez Kernel Function belongs to the Smirnov class, $E^q(D)$, for every $q\\in(0,N)$.
The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\\subset\\mathbb{C}$ of class $C^2$ if we fix $\\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function
\\mathcal{K}(\\zeta,z)=\\frac{1}{2\\pi i}\\frac{1}{\\zeta-z}
as a function of $z$, then the Cauchy Kernel Function belongs to the Smirnov class $E^q(D)$ for every $q\\in(0,1)$.
"]},{"key":"dc:title","label":"Title","values":["Hardy Space Properties of the Cauchy Kernel Function for a Strictly Convex Planar Domain"]}]}],"canonical_facts":{"dc:contributor":["Harrington, Phillip S.","Luecking, Daniel H."],"dc:contributor.advisor":["Lanzani, Loredana"],"dc:creator":["Espinosa Lucio, Belen"],"dc:date":["2013"],"dc:date.available":["2017-09-29T07:00:00Z"],"dc:description.abstract":["This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class $C^2$ in $\\mathbb{C}^N$, the Henkin-Ram\\'irez Kernel Function belongs to the Smirnov class, $E^q(D)$, for every $q\\in(0,N)$.
The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\\subset\\mathbb{C}$ of class $C^2$ if we fix $\\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function
\\mathcal{K}(\\zeta,z)=\\frac{1}{2\\pi i}\\frac{1}{\\zeta-z}
as a function of $z$, then the Cauchy Kernel Function belongs to the Smirnov class $E^q(D)$ for every $q\\in(0,1)$.
"],"dc:identifier":["https://scholarworks.uark.edu/etd/766"],"dc:subject":["Pure sciences","Applied sciences","Cauchy kernel","Complex variables","Hardy spaces","Smirnov space","Numerical Analysis and Computation"],"dc:title":["Hardy Space Properties of the Cauchy Kernel Function for a Strictly Convex Planar Domain"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy in Mathematics (PhD)"]},"updated_at":"2026-07-24T00:59:32Z"}