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Spectral projections and resolvent estimates on Damek-Ricci spaces and their applications
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abstract
We prove $L^p-L^{p^\prime}$ boundedness of spectral projections and the resolvent of the Laplace-Beltrami operator on Damek-Ricci spaces with the explicit norms in terms of the spectral parameter. To prove these results we established pointwise sharp bounds on the spherical functions and their derivatives. As an application, we study the eigenvalue bounds of Schr\"odinger operators with complex valued potential.
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Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces
On Damek-Ricci spaces, radial initial data in H^β with β>a/4 give almost everywhere pointwise convergence for dispersive equations with asymptotically concave phase of degree a; the threshold is sharp up to the endpoint.
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