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Let $X$ be a metric space with doubling measure, and $L$ be a nonnegativeself-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussianupper bound. Let $f$ be in the space $ {\rm BMO}_L(X)$ associated with theoperator $L$ and we define its distance from the subspace $L^{\infty}(X)$ underthe $ {\rm BMO}_L(X)$ norm as follows: $$  {\rm dist} (f, L^{\infty}):= \inf_{g\in L^{\infty}} \|f -g\|_{{\rmBMO}_L(X)}. $$ In this paper we prove that ${\rm dist} (f, L^{\infty})$ isequivalent to the infimum of the constant $\varepsilon$ in the John-Nirenberginequality for the space ${\rm BMO}_L(X)$: $$  \sup_B { \mu\big(\{ x\in B: |f(x)-e^{-{r_B^2}L}f(x)|>\lambda\}\big) \over\mu(B)} \leq e^{-\lambda/\varepsilon}\ \ \ \ {\rm for\ large\ } \lambda. $$This extends the well-known result of Garnett and Jones \cite{GJ1} for theclassical ${\rm BMO}$ space (introduced by John and Nirenberg). As anapplication, we show that a ${\rm BMO}_L(X)$ function with compact support canbe decomposed as the summation of an $L^\infty$-function and the integral ofthe heat kernel (associated with $L$) against a finite Carleson measure on$X\times[0,\infty)$. The key new technique is a geometric constructioninvolving the semigroup $e^{-tL}$. We also resort to several fundamental toolsincluding the stopping time argument and the random dyadic lattice.

The Garnett-Jones Theorem on BMO spaces associated with operators and applications