On Dunkl Schr"odinger semigroups with Green bounded potentials (original) (raw)

On R N equipped with a normalized root system R, a multiplicity function k(α) > 0, and the associated measure dw(x) = α∈R | x, α | k(α) dx, we consider a Dunkl Schrödinger operator L = −∆ k + V , where ∆ k is the Dunkl Laplace operator and V ∈ L 1 loc (dw) is a non-negative potential. Let h t (x, y) and k {V } t (x, y) denote the Dunkl heat kernel and the integral kernel of the semigroup generated by −L respectively. We prove that k {V } t (x, y) satisfies the following heat kernel lower bounds: there are constants C, c > 0 such that h ct (x, y) ≤ Ck {V } t (x, y) if and only if sup x∈R N ∞ 0 R N V (y)w(B(x, √ t)) −1 e − x−y 2 /t dw(y) dt < ∞, where B(x, √ t) stands for the Euclidean ball centered at x ∈ R N and radius √ t.