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 $1 < p < \infty$.
Categories: math.AP math.CA