  Consider a non-doubling manifold with ends $M = \mathfrak{R}^{n}\sharp\,
{\mathbb R}^{m}$ where $\mathfrak{R}^n=\mathbb{R}^n\times \mathbb{S}^{m-n}$ for
$m> n \ge 3$. We say that an operator $L$ has a generalised Poisson kernel if
$\sqrt{ L}$ generates a semigroup $e^{-t\sqrt{L}}$ whose kernel $p_t(x,y)$ has
an upper bound similar to the kernel of $e^{-t\sqrt{\Delta}}$ where $\Delta$ is
the Laplace-Beltrami operator on $M$. An example for operators with generalised
Gaussian bounds is the Schr\"odinger operator $L = \Delta + V$ where $V$ is an
arbitrary non-negative locally integrable potential. In this paper, our aim is
to introduce the BMO space ${\rm BMO}_L(M)$ associated to operators with
generalised Poisson bounds which serves as an appropriate setting for certain
singular integrals with rough kernels to be bounded from $L^{\infty}(M)$ into
this new ${\rm BMO}_L(M)$. On our ${\rm BMO}_L(M)$ spaces, we show that the
John--Nirenberg inequality holds and we show an interpolation theorem for a
holomorphic family of operators which interpolates between $L^q(M)$ and ${\rm
BMO}_L(M)$. As an application, we show that the holomorphic functional calculus
$m(\sqrt{L})$ is bounded from $L^{\infty}(M)$ into ${\rm BMO}_L(M)$, and
bounded on $L^p(M)$ for $1 < p < \infty$.
