Spectral multipliers, Bochner-Riesz means and uniform Sobolev inequalities for elliptic operators

This paper comprises two parts. In the first, we study $L^p$ to $L^q$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators $P(D)-z$ and all $z\in {\mathbb C}\backslash [0, \infty)$, which give an extension of the second order results of Kenig-Ruiz-Sogge \cite{KRS}. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schr\"odinger operators $P(D)+V$ with small integrable potentials $V$. Finally we deduce spectral multiplier estimates for all these operators, including sharp Bochner-Riesz summability results.