Weighted estimates for the area integral associated with self-adjoint operators on homogeneous space

Let X be a metric space with doubling measure, and L be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy Gaussian bounds. This article is concerned with some weighted norm inequalities for area integrals associated with L . As an application, we obtain sharp estimates for the operator norm of the area integrals on L p ( X ) as p becomes large.