부산대학교 도서관

This article is a survey of Littlewood-Paley theory for semigroupsacting 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$$Gf(\theta)=\left(\int^1_0(1-r)^2\Vert\nabla P_r\astf(\theta)\Vert^2{dr\over1-r}\right)^{1/2}$$where$$\Vert\nabla P_r\ast f(\theta)\Vert=\left(\left|{\partial P_r\over\partialr}\ast f(\theta)\right|^2+\left|{1\over r}{\partial P_r\over\partial\theta}\astf(\theta)\right|^2\right)^{1/2}.$$A classical fact is that for any $p\in(1,\infty)$, there exists aconstant $C$ 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 generalizedLittlewood-Paley $g$-function is defined as$$G_qf(\theta)=\left(\int^1_0(1-r)^q\Vert\nabla P_r\astf(\theta)\Vert^q_{\scr B}{dr\over1-r}\right)^{1/q}.$$A Banach space $\scr B$ is said to be of Lusin cotype $q$ if thereexist $p\in(1,\infty)$ 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 cotypeproperty by using semigroup theory.