2016/12/13 by Christoph Kriegler, Kriegler, Christoph, Lutz Weis +1
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1612.04142
openalex publication_date 2016/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove spectral multiplier theorems for Hörmander classes Hα_p for 0-sectorial operators A on Banach spaces assuming a bounded H^∞(Σ_σ) calculus for some σ∈ (0,π) and norm and certain R-bounds on one of the following families of operators: the semigroup e--zA on ℂ_+, the wave operators eisA for s ∈ ℝ, the resolvent (λ-- A)-1 on ℂ \backslash ℝ, the imaginary powers Ait for t ∈ ℝ or the Bochner-Riesz means (1-A/u)α_+ for u > 0. In contrast to the existing literature we neither assume that A operates on an Lp scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) Hα_1 calculus in terms of R-boundedness of Bochner-Riesz means.