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

Consider a non-doubling manifold with ends M=Rⁿ♯R^m where Rⁿ=Rⁿ×S^m−n for m>n≥3. We say that an operator L has a generalised Poisson kernel if L generates a semigroup e^−tL whose kernel p_t(x,y) has an upper bound similar to the kernel of e^−tΔ where Δ is the Laplace-Beltrami operator on M. An example for operators with generalised Gaussian bounds is the Schrödinger operator L=Δ+V where V is an arbitrary non-negative locally integrable potential. In this paper, our aim is to introduce the BMO space 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^∞(M) into this new BMO_L(M). On our 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 BMO_L(M). As an application, we show that the holomorphic functional calculus m(L) is bounded from L^∞(M) into BMO_L(M), and bounded on L^p(M) for 1<p<∞.