Remarks on Spectral Multiplier Theorems on Hardy Spaces Associated with Semigroups of Operators

sup t>0 ‖η( · )m(t · )‖W 2,β(Rd) ≤ Cη, where ‖ · ‖W 2,β(Rd) is the standard Sobolev norm on R, then the multiplier operator f 7→ F−1(mFf), initially defined on L(R) ∩ L(R), is bounded on L(R) for 1 < p < ∞, and is of weak-type (1,1). Here F denotes the Fourier transform. Let (Ω, d(x, y)) be a metric space equipped with a positive measure μ. We assume that (Ω, d, μ) is a space of homogeneous type in the sense of CoifmanWeiss [9], that is, there exists a constant C > 0 such that μ(Bd(x, 2t)) ≤ Cμ(Bd(x, t)) for every x ∈ Ω, t > 0, (1.1) where Bd(x, t) = {y ∈ Ω : d(x, y) < t}. The condition (1.1) implies that there exist constants C > 0 and q > 0 such that μ(Bd(x, st)) ≤ C0sμ(Bd(x, t)) for every x ∈ Ω, t > 0, s > 1. (1.2)