Spectral multipliers for Laplacians associated with some Dirichlet forms

Abstract

Let ℓ =-Δ-ℓ-1∇ℓ · ∇ be the self-adjoint operator associated with the Dirichlet form equation presented where ℓ is a positive C2 function, dλℓ = ℓdλ and λ denotes Lebesgue measure on ℝd. We study the boundedness on Lp(λ ℓ) of spectral multipliers of ^\varphi$. We prove that if ℓ grows or decays at most exponentially at infinity and satisfies a suitable curvature condition, then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin-type conditions at infinity are spectral multipliers of Lp(λ ℓ). The parabolic region depends on ℓ, on p and on the infimum of the essential spectrum of the operator $\smash{\mathcal{L}^\ varphi}$ on L2(λℓ). The sector depends on the angle of holomorphy of the semigroup generated by ℓ on Lp(λℓ). © 2008 Copyright Edinburgh Mathematical Society.