Spectral projections and resolvent estimates on Damek–Ricci spaces and their applications

Abstract We prove <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>-</m:mo> <m:msup> <m:mi>L</m:mi> <m:msup> <m:mi>p</m:mi> <m:mo>′</m:mo> </m:msup> </m:msup> </m:mrow> </m:math> {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 Schrödinger operators with complex valued potential.