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arXiv · 2605.30766

Two-weight inequalities for the Dunkl--Poisson integrals

Qingdong Guo·Ji Li·Brett D. Wick·Liangchuan Wu
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Abstract

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.

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Qingdong Guo, Ji Li, Brett D. Wick, Liangchuan Wu. 2026-05-29. Two-weight inequalities for the Dunkl--Poisson integrals. https://arxiv.org/abs/2605.30766

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