Due to the non‐commutativity of quaternions, there are three kinds of quaternionic Banach spaces, i.e. the right sided, the left sided, and the two sided ones. For the left and right sided Banach spaces, the quaternion multiplications are only defined on one single side. In this paper, we characterize the quaternionic Banach spaces such that the quaternionic operator valued Fourier transforms satisfy the Hausdorff‐Young inequalities. Then, with these Banach spaces, the vector‐valued Pitt inequalities are generalized to the quaternion setting. We also show that the Mikhlin type multiplier theorems can be transplanted to this setting. All these results are obtained based on the relationship between the complex operator‐valued Fourier transform and the quaternionic operator‐valued Fourier transforms.

中文翻译：

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四元数算子值傅里叶变换的Hausdorff-Young不等式和乘子定理

由于四元数的非可交换性，四元数Banach空间存在三种，即右侧，左侧和两侧。对于左侧和右侧的Banach空间，四元数乘法仅在一侧定义。在本文中，我们对四元离子Banach空间进行了刻画，以使四元离子算符值的Fourier变换满足Hausdorff-Young不等式。然后，使用这些Banach空间，将向量值的Pitt不等式推广到四元数设置。我们还表明，可以将Mikhlin型乘数定理移植到此设置中。所有这些结果都是基于复杂算子值傅里叶变换和四元数算子值傅里叶变换之间的关系获得的。