论文标题
Littlewood-Paley-Stein功能：一种R型方法
Littlewood-Paley-Stein functionals: an R-boundedness approach
论文作者
论文摘要
令$ l =δ+ v $是一个完全riemannian歧管$ m $的schrödinger运营商，具有非负势$ v $。我们证明，与$ l $相关的垂直Littlewood-paley-Stein功能在$ l^p（m）$ {\ it上且仅在且仅考虑}集合$ \ {\ sqrt {\ sqrt {t} \，\ nabla e^{ - tl} { - tl}，\ tl}，\，\，\，t> 0 \} $ liff Fiff the我们还介绍和研究更多的一般功能。对于函数序列$ m_k：[0，\ infty）\ to \ mathbb {c} $，我们定义$$ h（（（f_k））：= \ big（\ sum_k \ sum_0^\ int_0^\ infty | \ nabla m_k（tl）m_k（tl）m_k（tl） \ int_0^\ infty |。 h（（f_k））\ | _p \ le c \，\ big \ | \ big（\ sum_k | f_k |^2 \ big）^{1/2} \ big \ | _p $$对于某些常数$ c $独立于$（f_k）_k $。在双空间$ l^{p'} $上也证明了较低的估计值。我们介绍和研究其他Littlewood-Paley-Stein型功能的界限，并讨论它们与Riesz变换的关系。论文中给出了几个例子。
Let $L = Δ+ V$ 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 $$H((f_k)) := \Big( \sum_k \int_0^\infty | \nabla m_k(tL) f_k |^2 dt \Big)^{1/2} + \Big( \sum_k \int_0^\infty | \sqrt{V} m_k(tL) f_k |^2 dt \Big)^{1/2}.$$ Under fairly reasonable assumptions on $M$ we prove boundedness of $H$ on $L^p(M)$ in the sense $$\| H((f_k)) \|_p \le C\, \Big\| \Big( \sum_k |f_k|^2 \Big)^{1/2} \Big\|_p$$ for some constant $C$ independent of $(f_k)_k$. A lower estimate is also proved on the dual space $L^{p'}$. 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.
