Proper cocycles, measure equivalence and L_p-Fourier multipliers

2025/06/23 by Wang, Simeng, Xia, Runlian, Yao, Gan
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2506.18320

Abstract

We develop a new transference method for completely bounded L_p-Fourier multipliers via proper cocycles arising from probability measure-preserving group actions. This method extends earlier results by Haagerup and Jolissaint, which were limited to the case p = ∞. Based on this approach, we present a new and simple proof of the main result in [Hong-Wang-Wang, Mem. Amer. Math. Soc. 2024] regarding the pointwise convergence of noncommutative Fourier series on amenable groups, refining the associated estimate for maximal inequalities. In addition, this framework yields a transference principle for L_p-Fourier multipliers from lattices in linear Lie groups to their ambient groups, establishing a noncommutative analogue of Jodeit's theorem. As a further application, we construct a natural analogue of the Hilbert transform on SL₂(ℝ).

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