$L^p$ estimates for Dunkl--Schrödinger operators

We study the Dunkl–Schrödinger operator $L=-{\mathrm{\Delta }}_k+V$ on ${\mathbb{R}}^N$, $N>2$, where $V$ is a nonnegative potential in the reverse Hölder class ${\mathrm{R}\mathrm{H}}^q(dw)$, $q>\mathbf{N}\mathrm{/}2$, and $\mathbf{N}$ is the homogeneous dimension. We establish decay estimates for the resolvent kernel in terms of the critical radius and the orbit distance. For reflection-invariant potentials, we obtain $L^p(dw)$ bounds for $V{L}^{-1}$, second-order Dunkl derivatives of $L^{-1}$, and the Riesz transform ${\mathrm{\nabla }}_k{L}^{-1\mathrm{/}2}$, together with fractional potential and mixed estimates. The proofs combine local estimates for weak solutions, the Dunkl Fefferman–Phong inequality, and kernel estimates adapted to the Euclidean and orbit distances.