AI 中文总结
本文引入抽象各向异性权矩阵的卷积，研究其性质，将其应用于Braun-Meise-Taylor权函数相关的各向同性矩阵，最终用于处理双曲型偏微分方程正则性受控损失下的局部可解性问题。
AI 中文摘要
我们引入了抽象给定的各向异性权矩阵之间的卷积，并研究了这一新构造的性质。进一步，我们将相关知识应用于特定但有趣的情形：即（各向同性）矩阵与Braun-Meise-Taylor意义下的权函数相关联，并展示了相关权矩阵的卷积如何修改基础权函数。最后，我们给出了该卷积在研究某类双曲型偏微分方程局部可解性概念时的具体应用。实际上，该卷积可自然处理混合情形，即通过两个通常不同的权序列表达的正则性受控损失。
英文摘要
We introduce the convolution between abstractly given anisotropic weight matrices and investigate properties of this new construction. Further, we apply the knowledge to the particular but interesting case when the (isotropic) matrices are associated with weight functions in the sense of Braun-Meise-Taylor and show how the convolution of associated weight matrices modifies the underlying weight functions. Finally, we give a concrete application of the convolution when studying the concept of local solvability for a certain hyperbolic PDE. Indeed, the convolution allows to treat in a natural way a mixed setting; i.e. having a controlled loss of regularity expressed in terms of two, in general different, weight sequences.
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