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

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Coulhon T., Duong X., Li X.

Studia Mathematica

2003

tom 154

nr 1

37-57

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.

Ograniczanie wyników

1 Studia Mathematica

1 Coulhon T.

1 Duong X.

1 Li X.

1 2003

Wyniki wyszukiwania

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