Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group
The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type <inline-formula><math xmlns="http://...
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2024-10-01
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| author | Jinsen Xiao Jianxun He Yingzhu Wu |
| author_facet | Jinsen Xiao Jianxun He Yingzhu Wu |
| author_sort | Jinsen Xiao |
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| description | The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type <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> multiplier theorem on the Heisenberg group. If an operator-valued function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfies certain conditions, the right-multiplier operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math></inline-formula> is bounded on the Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">H</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems. |
| format | Article |
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| language | English |
| publishDate | 2024-10-01 |
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| spelling | doaj-art-da0203e4d21d4e8eaf80c5c4e2df207a2025-08-20T01:53:40ZengMDPI AGAxioms2075-16802024-10-01131174510.3390/axioms13110745Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg GroupJinsen Xiao0Jianxun He1Yingzhu Wu2Department of Mathematics, Guangdong University of Petrochemical Technology, Maoming 525000, ChinaDepartment of Basic Course Teaching, Software Engineering Institute of Guangzhou, Guangzhou 510990, ChinaDepartment of Mathematics, Guangdong University of Petrochemical Technology, Maoming 525000, ChinaThe classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type <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> multiplier theorem on the Heisenberg group. If an operator-valued function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfies certain conditions, the right-multiplier operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math></inline-formula> is bounded on the Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">H</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems.https://www.mdpi.com/2075-1680/13/11/745multiplierHeisenberg groupHardy spaceFourier transform |
| spellingShingle | Jinsen Xiao Jianxun He Yingzhu Wu Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group Axioms multiplier Heisenberg group Hardy space Fourier transform |
| title | Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group |
| title_full | Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group |
| title_fullStr | Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group |
| title_full_unstemmed | Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group |
| title_short | Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group |
| title_sort | mikhlin type i h sup p sup i multiplier theorem on the heisenberg group |
| topic | multiplier Heisenberg group Hardy space Fourier transform |
| url | https://www.mdpi.com/2075-1680/13/11/745 |
| work_keys_str_mv | AT jinsenxiao mikhlintypeihsuppsupimultipliertheoremontheheisenberggroup AT jianxunhe mikhlintypeihsuppsupimultipliertheoremontheheisenberggroup AT yingzhuwu mikhlintypeihsuppsupimultipliertheoremontheheisenberggroup |