arXiv:2611.12984v1 Announce Type: new Abstract: Let \(\Lk=-\Dk\) denote the nonnegative Dunkl Laplacian associated with a finite reflection group and a nonnegative multiplicity function \(\kappa\), and let $ N=d+2\gk\geq 3 $ be the corresponding homogeneous dimension. We establish uniform \(L_\kappa^p(\mathbb{R}^d)\)-\(L_\kappa^q(\mathbb{R}^d)\) resolvent estimates for \(\Lk\) whenever \[ \frac{2}{N+1} \leq \frac{1}{p}-\frac{1}{q} \leq \frac{2}{N}, \quad \frac{2N}{N+3}<p<\frac{2N}{N+1}, \quad \frac{2N}{N-1}<q<\frac{2N}{N-3}, \] where the last upper bound is interpreted as \(q<\infty\) when \(N=3\). Remarkably, this admissible range coincides with the corresponding range for the Euclidean Laplacian \(-\Delta\) in dimension \(N\).
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