### Abstract

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 ℝ^{n}.

Original language | English |
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Pages (from-to) | 37-57 |

Number of pages | 21 |

Journal | Studia Mathematica |

Volume | 154 |

Issue number | 1 |

Publication status | Published - 2003 |

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## Cite this

*Studia Mathematica*,

*154*(1), 37-57.