Abstract
In this paper we consider the Laplace–Beltrami operator Δ on Damek–Ricci spaces and derive pointwise estimates for the kernel of e τΔ, when τ ∈ ℂ* with Re τ ≥0. When τ ∈iℝ*, we obtain in particular pointwise estimates of the Schrödinger kernel associated with Δ. We then prove Strichartz estimates for the Schrödinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice [4] on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schrödinger equation associated with a distinguished Laplacian on Damek–Ricci spaces, showing that in this case the standard L 1 → L ∞ estimate fails while suitable weighted Strichartz estimates hold.
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