Let L be a non-negative self adjoint operator on L2(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 Lp(X) for 1 < p < ∞ when F is a suitable function.
The Anh BUI. "Commutators of BMO functions with spectral multiplier operators." J. Math. Soc. Japan 64 (3) 885 - 902, July, 2012. https://doi.org/10.2969/jmsj/06430885