We calculate the norm of the Fourier operator from \(L^p(X)\) to \(L^q({\hat{X}})\) when X is an infinite locally compact abelian group that is, furthermore, compact or discrete. This subsumes the sharp Hausdorff–Young inequality on such groups. In particular, we identify the region in (p, q)-space where the norm is infinite, generalizing a result of Fournier, and setting up a contrast with the case of finite abelian groups, where the norm was determined by Gilbert and Rzeszotnik. As an application, uncertainty principles on such groups expressed in terms of Rényi entropies are discussed.

中文翻译：

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紧或离散阿贝尔群的傅立叶变换的范数

当X是无限局部紧致的阿贝尔群时，我们计算从\（L ^ p（X）\）到\（L ^ q（{\ hat {X}}）\）的傅立叶算子的范数。紧凑或离散。这归因于这类群体的Hausdorff-Young严重不平等。特别是，我们确定（p，  q）空间中范数是无限的区域，推广了Fournier的结果，并与由Gilbert和Rzeszotnik确定范数的有限阿贝尔群的情况形成对比。作为一种应用，讨论了用Rényi熵表示的此类群的不确定性原理。