Spectral Multipliers, Bochner–Riesz Means and Uniform Sobolev Inequalities for Elliptic Operators

Abstract

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 |$z\in {\mathbb C}\backslash [0, \infty)$|⁠, 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.