Spectral multipliers for wave operators

2012/10/16 by Christoph Kriegler, Kriegler, Christoph
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1210.4261

arxiv created 2012/10/16 · openalex publication_date 2012/10/16 · arxiv updated 2012/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A classical theorem of Mihlin yields Lp estimates for spectral multipliers Lp(R^d) -> Lp(R^d); g -> F^-1[f(| |²) Fg] in terms of L^∞ bounds of the multiplier function f and its weighted derivatives up to an order > d/2. This theorem, which is a functional calculus for the standard Laplace operator, has generalisations in several contexts such as elliptic operators on domains and manifolds, Schrödinger operators and sublaplacians on Lie groups. However, for the wave equation functions f (s) = (1 + s)^-α e^its a better estimate is available, in the standard case (works of Miyachi and Peral) and on Heisenberg Lie groups (Müller and Stein). By a transference method for polynomially bounded regularized groups, we obtain a new class of spectral multipliers for operators that have these better wave spectral multipliers and that admit a spectral decomposition of Paley-Littlewood type.

Related

• Oscillating spectral multipliers on groups of Heisenberg type
• Spectral multiplier theorems via H^∞ calculus and R-bounds
• Spectral multipliers via resolvent type estimates on non-homogeneous metric measure spaces
• Spectral multipliers on Métivier groups
• L^p-spectral multipliers for some elliptic systems