In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As this specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.

中文翻译：

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复合算子在Morrey空间和弱Morrey空间上的有界性

在这项研究中，我们研究了作用于Morrey空间和弱Morrey空间上的合成算子的有界性。这项研究的主要目的是研究由Morrey空间上的一个同构引起的合成算子的有界性的充要条件。特别地，详细信息是从有界性中得出的，即，导致合成操作符的映射的双里普希茨连续性是从合成映射的连续性中得出的。证明的思想是确定特征函数的Morrey范数，并采用由特征函数组成的特定函数。由于此特定函数属于Morrey空间但不属于Lebesgue空间，因此结果揭示了Lebesgue空间中未观察到的新现象。随后，我们证明了映射满足组合空间算子的有界性，该映射满足由赋范空间生成的一般弱类型空间的适当体积估计。作为推论，提供了在弱的Morrey空间上合成算子的有界性的充要条件。