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

Abstract

Consider a non-doubling manifold with ends M=Rn♯Rm where Rn=Rn×Sm−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 pt(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 BMOL(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 BMOL(M). On our BMOL(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 Lq(M) and BMOL(M). As an application, we show that the holomorphic functional calculus m(L) is bounded from L∞(M) into BMOL(M), and bounded on Lp(M) for 1<p<∞.