Mikhlin-Type <i>H<sup>p</sup></i> Multiplier Theorem on the Heisenberg Group

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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Bibliographic Details
Main Authors: Jinsen Xiao, Jianxun He, Yingzhu Wu
Format: Article
Language:English
Published: MDPI AG 2024-10-01
Series:Axioms
Subjects:
multiplier
Heisenberg group
Hardy space
Fourier transform
Online Access:https://www.mdpi.com/2075-1680/13/11/745
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Summary: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.
ISSN:2075-1680

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