부산대학교 도서관

This article is a survey of Littlewood-Paley theory for semigroups acting on $L^p$-spaces of functions with values in a Banach space $\scr B$. Let $T=[-\pi,\pi]$ and $$ P_r(\theta)={1\over2\pi}{1-r\over1+r^2-2r\cos\theta} $$ be the Poisson kernel for the disc. The classical Littlewood-Paley $g$-function is defined, for $f\in L^p(T),\ 1\leq p\leq\infty$, as $$ Gf(\theta)=\left(\int^1_0(1-r)^2\Vert\nabla P_r\ast f(\theta)\Vert^2{dr\over1-r}\right)^{1/2} $$ where $$ \Vert\nabla P_r\ast f(\theta)\Vert=\left(\left|{\partial P_r\over\partial r}\ast f(\theta)\right|^2+\left|{1\over r}{\partial P_r\over\partial\theta}\ast f(\theta)\right|^2\right)^{1/2}. $$ A classical fact is that for any $p\in(1,\infty)$, there exists a constant $C$ depending only on $p$ such that $$ \Vert Gf\Vert_{L^p(T)}\leq C\Vert f\Vert_{L^p(T)}. $$ For $q\in(1,\infty)$ and $f\in L^1_{\scr B}(T)$, the generalized Littlewood-Paley $g$-function is defined as $$ G_qf(\theta)=\left(\int^1_0(1-r)^q\Vert\nabla P_r\ast f(\theta)\Vert^q_{\scr B}{dr\over1-r}\right)^{1/q}. $$ A Banach space $\scr B$ is said to be of Lusin cotype $q$ if there exist $p\in(1,\infty)$ and a positive constant $C$ such that $$ \Vert G_qf\Vert_{L^p(T)}\leq C\Vert f\Vert_{L^p_{\scr B}(T)}. $$ In this paper the author gives characterizations of the Lusin cotype property by using semigroup theory.