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

Abstract

We study the Dunkl--Schrödinger operator $L=-{\Delta }_k+V$ on ${\mathbb{R}}^N$, $N>2$, where $V$ is a nonnegative potential in the reverse Hölder class ${\mathrm{RH}}^q(dw)$, $q>\mathbf{N}/2$, 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 $V{L}^{-1}$, second-order Dunkl derivatives of $L^{-1}$, and the Riesz transform ${\nabla }_k{L}^{-1/2}$, 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.