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Hausdorff‐Young inequalities and multiplier theorems for quaternionic operator‐valued Fourier transforms
Author(s) -
Lian Pan
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7211
Subject(s) - mathematics , quaternion , multiplier (economics) , pure mathematics , commutative property , hausdorff space , fourier transform , banach space , mathematical analysis , algebra over a field , geometry , economics , macroeconomics
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.