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• # Artykuł - szczegóły

## Studia Mathematica

2003 | 154 | 1 | 37-57

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

EN

### Abstrakty

EN
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.

37-57

wydano
2003

### Twórcy

autor
• Université de Cergy-Pontoise, 95302 Pontoise, France
autor
• Macquarie University, North Ryde, NSW 2113, Australia
autor
• University of Oxford, Oxford OX1 3LB, United Kingdom