BMO spaces associated to operators with generalised Poisson bounds on non-doubling manifolds with ends

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 $1 < p < \infty$.