Upper and lower bounds for the Dunkl heat kernel

Abstract

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