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We study the Möbius invariant spaces$Q$_{$p$}and$Q$_{$p, 0$}of analytic functions. These scales of spaces include BMOA=$Q$_{1}, VMOA=$Q$_{1, 0}and the Dirichlet space=$Q$_{0}. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for both$Q$_{$p$}and$Q$_{$p, 0$}.

We prove that the linearized Riesz transforms and the imaginary powers of the Laplacian are$H$^{1}-bounded on complete Riemannian manifolds satisfying the doubling property and the Poincaré inequality, where$H$^{1}denotes the Hardy space on$M$.