Littlewood–Paley–Stein functions on complete Riemannian manifolds for $1\leq p\leq 2$
Volume 154 / 2003
Abstract
We study the weak type $(1, 1)$ and the $L^p$-boundedness, $1< p\le 2$, of the so-called vertical (i.e. involving space derivatives) Littlewood–Paley–Stein functions ${\cal G}$ and ${\cal H}$ respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold $M$. Without any assumption on $M$, we observe that ${\cal G}$ and ${\cal H}$ are bounded in $L^p$, $1< p\leq 2$. 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 ${\cal G}$ and ${\cal H}$ (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 ${\mathbb R}^n$.
