$L^p$ estimates for Dunkl--Schr\"odinger operators

Abstract

We study the Dunkl--Schr\"odinger operator $L=-\Delta_k+V$ on $\mathbb R^N$, $N>2$, where $V$ is a nonnegative potential in the reverse H\"older class ${\rm RH}^q(dw)$, $q>\mathbf N/2$, 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 $VL^{-1}$, second-order Dunkl derivatives of $L^{-1}$, and the Riesz transform $\nabla_kL^{-1/2}$, 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.