Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2

We study the weak type (1,1) and the L^p-boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions G and H, respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold M. Without any assumption on M, we observe that G and H are bounded in L^p, 1 < p ≤ 2. We also consider modified Littlewood-Paley-Stein functions that take into account the positivity of the bottom of the spectrum. Assuming that M satisfies the doubling volume property and an optimal on-diagonal heat kernel estimate, we prove that G and H (as well as the corresponding horizontal functions, i.e. involving time derivatives) are of weak type (1, 1). Finally, we apply our methods to divergence form operators on arbitrary domains of ℝⁿ.