Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. Let $f$ be in the space $ { \rm BMO}_L(X)$ associated with the operator $L$ and we define its distance from the subspace $L^{ \infty}(X)$ under the $ { \rm BMO}_L(X)$ norm as follows: $$ { \rm dist} (f, L^{ \infty}):= \inf_{ g\in L^{ \infty}} \|f -g\|_{ { \rm BMO}_L(X)}. $$ In this paper we prove that ${ \rm dist} (f, L^{ \infty})$ is equivalent to the infimum of the constant $\varepsilon$ in the John-Nirenberg inequality 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 the classical ${ \rm BMO}$ space (introduced by John and Nirenberg). As an application, we show that a ${ \rm BMO}_L(X)$ function with compact support can be decomposed as the summation of an $L^\infty$-function and the integral of the heat kernel (associated with $L$) against a finite Carleson measure on $X\times[0,\infty)$. The key new technique is a geometric construction involving the semigroup $e^{ -tL}$. We also resort to several fundamental tools including the stopping time argument and the random dyadic lattice.
Categories: math.CA math.AP