### The Garnett-Jones Theorem on BMO space associated with operators

Abstract

Let $BMO_L$ be the BMO space associated with an operator $L$. In this talk, we show two kinds of decomposition of $BMO_L$ functions. As an application, we prove the Garnett-Jones Theorem on $BMO_L$, that is, comparable upper and lower bounds are given for the distance for a $BMO_L$ function from $L^\infty$ space. The other decomposition claims that a compact supported $BMO_L$ function can be decomposed as the summation of an $L^\infty$ function and the integral of the heat kernel with respect to a finite Carleson measure. Random dyadic lattice method is used in both of the two kinds of decomposition. All condition we need is the heat kernel of the semigroup generated by $L$ satisfies the Gaussian upper bound.