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By a new method derived from Nicola--Primo--Tabacco[24], we study theboundedness on $\alpha$-modulation spaces of unimodular multipliers with symbol$e^{i\mu(\xi)}$. Comparing with the previous results, the boundedness result isestablished for a larger family of unimodular multipliers under weakerassumptions.

Unimodular multipliers on $\alpha$-modulation spaces: A revisit with new method under weaker conditions