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Hardy's inequalities for the twisted convolution with Laguerre functions
1School of Sciences, Guangdong University of Petrochemical Technology, Maoming, 525000 P.R. China.
This study investigates Hardy
Area of Science:
- Harmonic Analysis
- Functional Analysis
- Special Functions
Background:
- Hardy's inequalities are fundamental in analysis.
- The twisted convolution and Laguerre functions present unique analytical challenges.
- The Heisenberg group provides a powerful framework for studying functions and operators.
Purpose of the Study:
- To establish new Hardy's inequalities involving the twisted convolution with Laguerre functions.
- To extend the understanding of inequalities in harmonic analysis.
- To explore the application of Heisenberg group methods in function space analysis.
Main Methods:
- Utilizing the Heisenberg group approach.
- Deriving estimates for Heisenberg left-invariant vectors of special Hermite functions.
- Applying these estimates to prove Hardy's inequalities.
Main Results:
- Two new types of Hardy's inequalities for the twisted convolution with Laguerre functions are proven.
- The effectiveness of the Heisenberg group method in this context is demonstrated.
- New bounds for special Hermite functions are established.
Conclusions:
- The study successfully establishes novel Hardy's inequalities.
- The Heisenberg group approach offers a viable and effective method for such problems.
- This work contributes to the theory of inequalities and special functions.
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