Upper and lower bounds for the Dunkl heat kernel

Abstract On $$\mathbb R^N$$ R N equipped with a normalized root system R , multiplicity function $$k(\alpha ) > 0$$ k ( α ) > 0 and the associated measure $$\begin{aligned} dw(\mathbf{x})=\prod _{\alpha \in R}|\langle \mathbf{x},\alpha \rangle |^{{k(\alpha )}}\, d\mathbf{x}, \end{aligned}$$ d w x = ∏ ∈ | ⟨ , ⟩ let $$h_t(\mathbf{x},\mathbf{y})$$ h t y denote heat kernel of semigroup generated by Dunkl Laplace operator $$\Delta _k$$ Δ . Let $$d(\mathbf{x},\mathbf{y})=\min _{{g}\in G} \Vert \mathbf{x}-{g}(\mathbf{y})\Vert $$ min g G ‖ - where G is reflection group We derive following upper lower bounds for : all $$c_l>1/4$$ c l 1 / 4 $$0<c_u<1/4$$ < u there are constants $$C_l,C_u>0$$ C such that C_{l}w(B(\mathbf {x},\sqrt{t}))^{-1}e^{-c_{l}\frac{d(\mathbf {x},\mathbf {y})^2}{t}} \Lambda (\mathbf{x},\mathbf{y},t) \le h_t(\mathbf {y}) C_{u}w(B(\mathbf {x},\sqrt{t}))^{-1}e^{-c_{u}\frac{d(\mathbf (\mathbf{x},\mathbf{y},t), B e 2 Λ ≤ $$\Lambda (\mathbf{x},\mathbf{y},t)$$ can be expressed means some rational functions $$\Vert /\sqrt{t}$$ An exact formula provided.