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

This article 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∈ℂ∖[0,∞), which give an extension of the second order results of Kenig–Ruiz–Sogge [42]. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schrödingier operators P(D)+V with small integrable potentials V. Finally, we deduce spectral multiplier applications for all these operators, including sharp Bochner–Riesz summability results.