论文标题
矢量值转移
Vector Valued Transference
论文作者
论文摘要
我们的主要结果是以下内容。 令$ x $和$ y $为Banach Space，让$ G $为本地紧凑的Abelian集团，让$ k $是在$ g $上定义的运算符，在$ g $上定义的，在有限线性运算符的空间中，从$ x $到$ y $。假设$ r $和$ \ tilde {r} $分别是$ x $和$ y $的$ g $的表示，将$ k $的值交织在一起。然后，在$ r，\ tilde {r} $和$ k $上的适当有限条件下，公式$$ t_kx = \ int_gk（u）r _ { - u} xdu $ $定义有界线性的线性运算符$ t_k $ t_k $从$ x $从$ x $ ty $ y $ knoctioned ty $ k $ k $ k l^$ k l^a $ k） $ l^p_y（g）$，（对于$ p $的所有值$ 1 \ le P <\ infty $。） 给出了Banach空间几何形状的许多应用。在抽象交换谐波分析的设置中证明了几个结果。我们概述了卢比奥·德·弗朗西亚（Rubio de Francia）的肯定决议的证据。这种转移技术用于获得$ \ bbb {r}^n $设置中某些操作员的无维度估计。
Our principal result is the following. Let $X$ and $Y$ be Banach spaces, let $G$ be a locally compact abelian group, and let $K$ be an operator valued kernel defined on $G$ with values in the space of bounded linear operators from $X$ to $Y$. Suppose that $R$ and $\tilde{R}$ are representations of $G$ on $X$ and $Y$ respectively that intertwine the values of $K$. Then, under suitable boundedness conditions on $R, \tilde{R}$ and $K$, the formula $$T_Kx = \int_GK(u)R_{-u}xdu $$ defines a bounded linear operator $T_K$ from $X$ to $Y$ with norm controlled by norm of convolution by $K$ as a mapping from $L^p_X(G)$ into $L^p_Y(G)$, (for all values of $p$ in the range $1\le p < \infty$.) A number of applications to the geometry of Banach spaces are given. Several results are proved in the setting of abstract commutative harmonic analysis. We outline the proof of the affirmative resolution of a conjecture of Rubio de Francia. This technique of transference is used to obtain dimension free estimates for certain operators in an $\Bbb{R}^n$ setting.
