Poisson Kernels and Hilbert Transforms for Trigonometric Heckman-Opdam Polynomials of type $A_1$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911318373564416 |
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| author | Amri, B. Guesmi, A. |
| author_facet | Amri, B. Guesmi, A. |
| contents | In this paper, we investigate the trigonometric Heckman-Opdam polynomials of type $A_1$. We establish connections with ultraspherical polynomials and derive an explicit expression for the associated Poisson kernel. Using the product formula, we introduce a natural convolution structure and develop a theory of fractional integrals associated with these polynomials. We also define a generalized Hilbert transform in the framework of the Cherednik operator and prove its boundedness on $L^p$-spaces.
This work provides an alternative perspective on the approach of B. Muckenhoupt and E.M. Stein \cite{MS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poisson Kernels and Hilbert Transforms for Trigonometric Heckman-Opdam Polynomials of type $A_1$ Amri, B. Guesmi, A. Classical Analysis and ODEs Analysis of PDEs 33C52, 42B10 In this paper, we investigate the trigonometric Heckman-Opdam polynomials of type $A_1$. We establish connections with ultraspherical polynomials and derive an explicit expression for the associated Poisson kernel. Using the product formula, we introduce a natural convolution structure and develop a theory of fractional integrals associated with these polynomials. We also define a generalized Hilbert transform in the framework of the Cherednik operator and prove its boundedness on $L^p$-spaces. This work provides an alternative perspective on the approach of B. Muckenhoupt and E.M. Stein \cite{MS}. |
| title | Poisson Kernels and Hilbert Transforms for Trigonometric Heckman-Opdam Polynomials of type $A_1$ |
| topic | Classical Analysis and ODEs Analysis of PDEs 33C52, 42B10 |
| url | https://arxiv.org/abs/2512.12659 |