Partial spectral multipliers and partial Riesz transforms for degenerate operators

Abstract

We consider degenerate differential operators of the type A = − d k,j=1 ∂k(akj∂j) on L2(Rd) with real symmetric bounded measurable coefficients. Given a function χ ∈ C ∞ b (Rd) (respectively, a bounded Lipschitz domain Ω), suppose that (akj) ≥ μ > 0 a.e. on supp χ (respectively, a.e. on Ω). We prove a spectral multiplier type result: if F : [0,∞) → C is such that supt>0 ϕ(.)F(t.) Cs d/2 then MχF(I + A)Mχ is weak type (1, 1) (respectively, 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) −1/2Mχ. The proofs are based on a criterion for a singular integral operator to be weak type (1, 1).