In this paper we consider the Laplace–Beltrami operator on Damek–Ricci spaces and derive pointwise estimates for the kernel of e^{\tau\Delta}, when \tau has nonnegative real part. We obtain in particular pointwise estimates of the Schrödinger kernel associated with \Delta. 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 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^\infty estimate fails while suitable weighted Strichartz estimates hold.

Schrödinger Equations on Damek–Ricci Spaces / Anker, J. P. h.; Pierfelice, V.; Vallarino, Maria. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - STAMPA. - 36:6(2011), pp. 976-997. [10.1080/03605302.2010.539658]

### Schrödinger Equations on Damek–Ricci Spaces

#### Abstract

In this paper we consider the Laplace–Beltrami operator on Damek–Ricci spaces and derive pointwise estimates for the kernel of e^{\tau\Delta}, when \tau has nonnegative real part. We obtain in particular pointwise estimates of the Schrödinger kernel associated with \Delta. 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 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^\infty estimate fails while suitable weighted Strichartz estimates hold.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2468179