Two-weight inequalities for the Dunkl--Poisson integrals
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910271600066560 |
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| author | Guo, Qingdong Li, Ji Wick, Brett D. Wu, Liangchuan |
| author_facet | Guo, Qingdong Li, Ji Wick, Brett D. Wu, Liangchuan |
| contents | We prove an $L^2$ two-weight testing theorem for the Dunkl--Poisson semigroup. The difficulty is geometric. The Dunkl orbit distance has several reflected diagonals, so a single orbit-box test may mix different chamber components. We avoid this by working on one Weyl chamber and keeping the chamber indices. Under the wall-null assumption the full operator becomes a finite matrix of scalar positive Poisson-type operators. In each entry the orbit diagonal is just the ordinary diagonal in the chamber variables. The scalar proof is then a principal-cube stopping-time argument, with two Dunkl kernel comparisons as the only new estimates. The resulting forward and backward tests are necessary and sufficient for the original Dunkl--Poisson inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30766 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two-weight inequalities for the Dunkl--Poisson integrals Guo, Qingdong Li, Ji Wick, Brett D. Wu, Liangchuan Classical Analysis and ODEs 42B20 We prove an $L^2$ two-weight testing theorem for the Dunkl--Poisson semigroup. The difficulty is geometric. The Dunkl orbit distance has several reflected diagonals, so a single orbit-box test may mix different chamber components. We avoid this by working on one Weyl chamber and keeping the chamber indices. Under the wall-null assumption the full operator becomes a finite matrix of scalar positive Poisson-type operators. In each entry the orbit diagonal is just the ordinary diagonal in the chamber variables. The scalar proof is then a principal-cube stopping-time argument, with two Dunkl kernel comparisons as the only new estimates. The resulting forward and backward tests are necessary and sufficient for the original Dunkl--Poisson inequality. |
| title | Two-weight inequalities for the Dunkl--Poisson integrals |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/2605.30766 |