The norm of the Fourier transform on compact or discrete abelian groups
Journal of Fourier Analysis and Applications
© 2020, Springer Science+Business Media, LLC, part of Springer Nature. We calculate the norm of the Fourier operator from Lp(X) to Lq(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.
Link to Published Version
Madiman, M., & Xu, P. (2020). The norm of the Fourier transform on compact or discrete abelian groups. Journal of Fourier Analysis and Applications, 26(3), 37. https://doi.org/10.1007/s00041-020-09737-7