Littlewood-Paley-Stein functionals: an R-boundedness approach

Abstract

Let $L=\Delta +V$ be a Schrödinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the vertical Littlewood-Paley-Stein functional associated with $L$ is bounded on $L^p(M)$ {\it if and only if} the set $\{\sqrt{t}\nabla e^{-tL},t>0\}$ is $\mathcal{R}$-bounded on $L^p(M)$. We also introduce and study more general functionals. For a sequence of functions $m_k:[0,\infty )\to \mathbb{C}$, we define $$H((f_k)):=(\sum _k{\int }_0^{\infty }|\nabla m_k(tL)f_k{|}^{2}dt{)}^{1/2}+(\sum _k{\int }_0^{\infty }|\sqrt{V}{m}_k(tL)f_k{|}^{2}dt{)}^{1/2}.$$ Under fairly reasonable assumptions on $M$ we prove boundedness of $H$ on $L^p(M)$ in the sense $$\Vert H((f_k)){\Vert }_p\le C\Vert (\sum _k|f_k{|}^2{)}^{1/2}{\Vert }_p$$ for some constant $C$ independent of $(f_k{)}_k$. A lower estimate is also proved on the dual space $L^{p^{\prime }}$. We introduce and study boundedness of other Littlewood-Paley-Stein type functionals and discuss their relationships to the Riesz transform. Several examples are given in the paper.

Improved version of Theorem 4.1 and several typos corrected. Final version to appear in Ann. Institut Fourier

Cited by in corpus (1)

• Conical square functionals on Riemannian manifolds