Paatero’s V(k) space II
In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2022-12-01
|
| Series: | Concrete Operators |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/conop-2022-0134 |
| _version_ | 1797947737203802112 |
|---|---|
| author | Andreev Valentin V. Bekker Miron B. Cima Joseph A. |
| author_facet | Andreev Valentin V. Bekker Miron B. Cima Joseph A. |
| author_sort | Andreev Valentin V. |
| collection | DOAJ |
| description | In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation. |
| first_indexed | 2024-04-10T21:31:30Z |
| format | Article |
| id | doaj.art-38d5811ff08f42e7994e20c57f7f6b59 |
| institution | Directory Open Access Journal |
| issn | 2299-3282 |
| language | English |
| last_indexed | 2024-04-10T21:31:30Z |
| publishDate | 2022-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Concrete Operators |
| spelling | doaj.art-38d5811ff08f42e7994e20c57f7f6b592023-01-19T13:20:29ZengDe GruyterConcrete Operators2299-32822022-12-019115115910.1515/conop-2022-0134Paatero’s V(k) space IIAndreev Valentin V.0Bekker Miron B.1Cima Joseph A.2Department of Mathematics, Lamar University, Beaumont, TX 77710Department of Mathematics, the University of Pittsburgh at Johnstown, 450 Schoolhouse, Johnstown, PA 15904Department of Mathematics, the University of North Carolina at Chapel Hill, CB 3250, 329 Phillips Hall, Chapel Hill, NC 27599In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.https://doi.org/10.1515/conop-2022-0134paatero classhp spacesgeometric function theory30c4530c1530h10 |
| spellingShingle | Andreev Valentin V. Bekker Miron B. Cima Joseph A. Paatero’s V(k) space II Concrete Operators paatero class hp spaces geometric function theory 30c45 30c15 30h10 |
| title | Paatero’s V(k) space II |
| title_full | Paatero’s V(k) space II |
| title_fullStr | Paatero’s V(k) space II |
| title_full_unstemmed | Paatero’s V(k) space II |
| title_short | Paatero’s V(k) space II |
| title_sort | paatero s v k space ii |
| topic | paatero class hp spaces geometric function theory 30c45 30c15 30h10 |
| url | https://doi.org/10.1515/conop-2022-0134 |
| work_keys_str_mv | AT andreevvalentinv paaterosvkspaceii AT bekkermironb paaterosvkspaceii AT cimajosepha paaterosvkspaceii |