Let Bn be the unit ball in Cn. If f is a bounded holomorphic function, we say that f is inner provided that its modulus has a radial limit of 1 almost everywhere on Sn where Sn is the unit sphere.
If a > -1 and p > 0 then Apa denotes the weighted Bergman space of all holomorphic functions weighted by (1-|z|^2)a and for 0 < q < 1, set Bq := A1(n/q)-n-1. For p > 0 let Hp denote the usual Hardy space of holomorphic functions on the ball.
In this dissertation, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, Rf(z) = (d/dt)f(tz)|t=1 will be considered in the Bp spaces for p > n/(n+1) and will be related to
membership in weighted Dirichlet spaces, weighted Bergman spaces A2a for 0 < a < 1, and to the Ap spaces for 1 < p < 2. Moreover, it will be shown that if f is an inner function, n > 1, and either Rf belongs to B2n/(2n+1) , A3/2, or H1/2 then f must be constant.
In addition to these results, we will also provide similar results for higher order derivatives as well.
Finally, we will briefly consider derivatives of invariant generalized Green potentials in the unit ball and examine the spaces to which they belong.
","abstract_html":"<p>Let Bn be the unit ball in Cn. If f is a bounded holomorphic function, we say that f is inner provided that its modulus has a radial limit of 1 almost everywhere on Sn where Sn is the unit sphere.</p> <p>If a > -1 and p > 0 then Apa denotes the weighted Bergman space of all holomorphic functions weighted by (1-|z|^2)a and for 0 < q < 1, set Bq := A1(n/q)-n-1. For p > 0 let Hp denote the usual Hardy space of holomorphic functions on the ball. </p> <p>In this dissertation, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, Rf(z) = (d/dt)f(tz)|t=1 will be considered in the Bp spaces for p > n/(n+1) and will be related to</p> <p> membership in weighted Dirichlet spaces, weighted Bergman spaces A2a for 0 < a < 1, and to the Ap spaces for 1 < p < 2. Moreover, it will be shown that if f is an inner function, n > 1, and either Rf belongs to B2n/(2n+1) , A3/2, or H1/2 then f must be constant. </p> <p>In addition to these results, we will also provide similar results for higher order derivatives as well. </p> <p>Finally, we will briefly consider derivatives of invariant generalized Green potentials in the unit ball and examine the spaces to which they belong.</p>","abstract_has_math":false,"creators":["Gamel, Matthew"],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Manfred Stoll"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-01-01T08:00:00Z","date_published":"2011-01-01T08:00:00Z","updated_at":"2026-07-24T04:37:34Z","subjects":["Mathematics","Physical Sciences and Mathematics","Inner function","M-Subharmonic function"],"languages":[],"rights":["© 2011, Matthew Gamel"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1595","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Manfred Stoll"]},{"key":"dc:creator","label":"Author","values":["Gamel, Matthew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics","Inner function","M-Subharmonic function"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2011, Matthew Gamel"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1595"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let Bn be the unit ball in Cn. If f is a bounded holomorphic function, we say that f is inner provided that its modulus has a radial limit of 1 almost everywhere on Sn where Sn is the unit sphere.
If a > -1 and p > 0 then Apa denotes the weighted Bergman space of all holomorphic functions weighted by (1-|z|^2)a and for 0 < q < 1, set Bq := A1(n/q)-n-1. For p > 0 let Hp denote the usual Hardy space of holomorphic functions on the ball.
In this dissertation, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, Rf(z) = (d/dt)f(tz)|t=1 will be considered in the Bp spaces for p > n/(n+1) and will be related to
membership in weighted Dirichlet spaces, weighted Bergman spaces A2a for 0 < a < 1, and to the Ap spaces for 1 < p < 2. Moreover, it will be shown that if f is an inner function, n > 1, and either Rf belongs to B2n/(2n+1) , A3/2, or H1/2 then f must be constant.
In addition to these results, we will also provide similar results for higher order derivatives as well.
Finally, we will briefly consider derivatives of invariant generalized Green potentials in the unit ball and examine the spaces to which they belong.
"]},{"key":"dc:title","label":"Title","values":["On the Integrability of Derivatives of Holomorphic and M-Subharmonic Inner Functions In the Unit Ball of Cn"]}]}],"canonical_facts":{"dc:contributor":["Manfred Stoll"],"dc:creator":["Gamel, Matthew"],"dc:description.abstract":["Let Bn be the unit ball in Cn. If f is a bounded holomorphic function, we say that f is inner provided that its modulus has a radial limit of 1 almost everywhere on Sn where Sn is the unit sphere.
If a > -1 and p > 0 then Apa denotes the weighted Bergman space of all holomorphic functions weighted by (1-|z|^2)a and for 0 < q < 1, set Bq := A1(n/q)-n-1. For p > 0 let Hp denote the usual Hardy space of holomorphic functions on the ball.
In this dissertation, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, Rf(z) = (d/dt)f(tz)|t=1 will be considered in the Bp spaces for p > n/(n+1) and will be related to
membership in weighted Dirichlet spaces, weighted Bergman spaces A2a for 0 < a < 1, and to the Ap spaces for 1 < p < 2. Moreover, it will be shown that if f is an inner function, n > 1, and either Rf belongs to B2n/(2n+1) , A3/2, or H1/2 then f must be constant.
In addition to these results, we will also provide similar results for higher order derivatives as well.
Finally, we will briefly consider derivatives of invariant generalized Green potentials in the unit ball and examine the spaces to which they belong.
"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1595"],"dc:rights":["© 2011, Matthew Gamel"],"dc:subject":["Mathematics","Physical Sciences and Mathematics","Inner function","M-Subharmonic function"],"dc:title":["On the Integrability of Derivatives of Holomorphic and M-Subharmonic Inner Functions In the Unit Ball of Cn"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T04:37:34Z"}