Commutators of BMO functions with spectral multiplier operators

Let L be a non-negative self adjoint operator on L ²(X) where X is a space of homogeneous type. Assume that L generates an analytic semigroup e- ^tL whose kernel satisfies the standard Gaussian upper bounds. By the spectral theory, we can define the spectral multiplier operator F(L). In this article, we show that the commutator of a BMO function with F(L) is bounded on L ^p(X) for 1 < p < ∞ when F is a suitable function.