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

2015/06/16 by Sikora, Adam, Yan, Lixin, Yao, Xiaohua
#58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.04883

Abstract

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∈ \mathbb C\backslash [0, ∞), which give an extension of the second order results of Kenig-Ruiz-Sogge \citeKRS. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schrödinger 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.

Related

• Spectral Multipliers II: Elliptic and Parabolic Operators and Bochner-Riesz Means
• Bounds on the maximal Bochner-Riesz means for elliptic operators
• Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means
• Bochner-Riesz means for the Hermite and special Hermite expansions
• Remarks on L^p-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems