Let \(\phi \) be an analytic map from the unit disk into itself, \(1<p<2\) and \(1\le q \le p\). It is shown that the composition operator \(C_\phi (f)=f\circ \phi \) is bounded from \(H^p_{weak}({\mathbb D},L^q(\mu ))\) into \(H^p({\mathbb D},L^q(\mu ))\) if and only if \(C_\phi \) is a 2-summing operator from \(H^p({\mathbb D})\) into \(H^p({\mathbb D})\). Here \(H^p_{weak}({\mathbb D},X)\) and \(H^p({\mathbb D},X)\) are the weak and strong formulation of X-valued Hardy spaces on the unit disc.

中文翻译：

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合成算子从弱到强向量值Hardy空间

假设\（\ phi \）是从单位磁盘到其本身\\（1 <p <2 \）和\（1 \ le q \ le p \）的解析图。证明了合成算子\（C_ \ phi（f）= f \ circ \ phi \）从\（H ^ p_ {weak}（{\ mathbb D}，L ^ q（\ mu））\ ）成\（H ^ p（{\ mathbb d}，L ^ q（\亩））\）当且仅当\（C_ \披\）是从2求和操作符\（H ^ p（{\ mathbb D}）\）放入\（H ^ p（{\ mathbb D}）\）中。这里\（H ^ p_ {weak}（{\ mathbb D}，X）\）和\（H ^ p（{\ mathbb D}，X）\）是X值Hardy空间的弱和强公式单位光盘。