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Littlewood-Paley inequality for a diffusion satisfying the logarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary

We discuss the Littlewood-Paley inequality for a diffusion process associated with a Dirichlet form of gradient type. We assume that the logarithmic Sobolev inequality holds and the negative part of Γ2 is exponentially integrable. Under these and some additional conditions, we showed that ‖∇u‖p ≤ C‖ √ 1− Lu‖q for 1 < p < q. We also discuss the Brownian motion on a Riemannian manifold with boundary.