2012/02/09 by A. F. M. ter Elst, ter Elst, A. F. M., El Maati Ouhabaz +1 · 1 citation
Computer Science · Mathematics · #42B15 #45F05 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems
paper · doi:10.48550/arxiv.1202.2136
openalex publication_date 2012/02/09 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We consider degenerate differential operators A = ∑k,j=1d ∂k (akj ∂j) on L2(ℝd) with real symmetric bounded measurable coefficients. Given a function χ∈ Cb^∞(ℝd) (respectively, Ω a bounded Lipschitz domain) and suppose that (akj) ≥ μ> 0 a.e. on \supp χ (resp., a.e. on Ω). We prove a spectral multiplier type result: if F\colon [0, ∞) → ℂ is such that supt > 0 ‖ φ(.) F(t .) ‖Cs < ∞ for some non-trivial function φ∈ Cc^∞(0,∞) and some s > d/2 then MχF(I+A) Mχ is weak type (1,1) (resp. PΩF(I+A) PΩ is weak type (1,1)). We also prove boundedness on Lp for all p ∈ (1,2] of the partial Riesz transforms Mχ∇ (I + A)-1/2M_ χ. The proofs are based on a criterion for a singular integral operator to be weak type (1,1).