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We give the sharp conditions for boundedness of Hausdorff operators on certain modulation and Wiener amalgam spaces.

In this paper, we consider the embedding relations between any two α-modulation spaces. Based on an observation that the α-modulation space with smaller α can be regarded as a corresponding α-modulation space with larger α, we give a complete characterization of the Fourier multipliers between α-modulation spaces with different α. Then we establish a full version of optimal embedding relations between α-modulation spaces.2000 Mathematics Subject Classification. 42B35.

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