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

Abstract Consider a non-doubling manifold with ends M = R n ♯ R m where R n = R n × S m − n for m > n ≥ 3 . We say that an operator L has a generalised Poisson kernel if L generates a semigroup e − t L whose kernel p t ( x , y ) has an upper bound similar to the kernel of e − t Δ where Δ is the Laplace-Beltrami operator on M. An example for operators with generalised Gaussian bounds is the Schrodinger 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 ∞ .