Generalized Vector-Valued Hardy Functions
We consider analytic functions in tubes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/2/39 |
| _version_ | 1797482744326914048 |
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| author | Richard D. Carmichael |
| author_facet | Richard D. Carmichael |
| author_sort | Richard D. Carmichael |
| collection | DOAJ |
| description | We consider analytic functions in tubes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mi>B</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> with values in Banach space or Hilbert space. The base of the tube <i>B</i> will be a proper open connected subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, an open connected cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, an open convex cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, and a regular cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mi>p</mi></msup></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></semantics></math></inline-formula>. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mi>p</mi></msup></semantics></math></inline-formula>. We obtain boundary value results on the distinguished boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mrow><mo>{</mo><mover><mn>0</mn><mo>¯</mo></mover><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and on the topological boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mo>∂</mo><mi>B</mi></mrow></semantics></math></inline-formula> of the tube for the analytic functions in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> and vector valued tempered distribution topologies. Suggestions for associated future research are given. |
| first_indexed | 2024-03-09T22:36:48Z |
| format | Article |
| id | doaj.art-869fc7f8ee564994ad9f5cc076fc5a55 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T22:36:48Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-869fc7f8ee564994ad9f5cc076fc5a552023-11-23T18:46:41ZengMDPI AGAxioms2075-16802022-01-011123910.3390/axioms11020039Generalized Vector-Valued Hardy FunctionsRichard D. Carmichael0Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, NC 27109, USAWe consider analytic functions in tubes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mi>B</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> with values in Banach space or Hilbert space. The base of the tube <i>B</i> will be a proper open connected subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, an open connected cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, an open convex cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, and a regular cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mi>p</mi></msup></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></semantics></math></inline-formula>. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mi>p</mi></msup></semantics></math></inline-formula>. We obtain boundary value results on the distinguished boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mrow><mo>{</mo><mover><mn>0</mn><mo>¯</mo></mover><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and on the topological boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mo>∂</mo><mi>B</mi></mrow></semantics></math></inline-formula> of the tube for the analytic functions in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> and vector valued tempered distribution topologies. Suggestions for associated future research are given.https://www.mdpi.com/2075-1680/11/2/39analytic functionsvector valued Hardy functionsboundary values |
| spellingShingle | Richard D. Carmichael Generalized Vector-Valued Hardy Functions Axioms analytic functions vector valued Hardy functions boundary values |
| title | Generalized Vector-Valued Hardy Functions |
| title_full | Generalized Vector-Valued Hardy Functions |
| title_fullStr | Generalized Vector-Valued Hardy Functions |
| title_full_unstemmed | Generalized Vector-Valued Hardy Functions |
| title_short | Generalized Vector-Valued Hardy Functions |
| title_sort | generalized vector valued hardy functions |
| topic | analytic functions vector valued Hardy functions boundary values |
| url | https://www.mdpi.com/2075-1680/11/2/39 |
| work_keys_str_mv | AT richarddcarmichael generalizedvectorvaluedhardyfunctions |