Abstract
Let G be a separable unimodular locally compact group of type I, and let N be a unimodular closed normal subgroup of G of type I, such that G/N is compact. Let for \(1<p\le 2\), \(\Vert {\mathscr {F}}^p(G)\Vert \) and \( \Vert {\mathscr {F}}^p(N )\Vert \) denote the norms of the corresponding \(L^p\)-Fourier transforms. We show that \(\Vert {\mathscr {F}}^p(G)\Vert \le \Vert {\mathscr {F}}^p(N )\Vert \). In the particular case where \(G=K < imes N\) is defined by a semi-direct product of a separable unimodular locally compact group N of type I and a compact subgroup K of the automorphism group of N, we show that equality holds if N has a K-invariant sequence \((\varphi _j)_j\) of functions in \(L^1(N)\cap L^p(N)\) such that \({\Vert {\mathscr {F}}\varphi _j \Vert _q}/{\Vert \varphi _j \Vert _p}\) tends to \(\Vert {\mathscr {F}}^p(N )\Vert \) when j goes to infinity. We show further that in some cases, an extremal function of N extends to an extremal function of G.
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The authors would like to thank the two Referees for having suggested many valuable comments to improve the final form of the paper.
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Communicated by Hartmut Führ.
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This work was completed with the support of D.G.R.S.R.T through the Research Laboratory LR 11ES52. This work was partially supported by JSPS KAKENHI Grant Numbers 25400115, JP17K05280.
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Baklouti, A., Inoue, J. The \(L^p\)-Fourier Transform Norm on Compact Extensions of Locally Compact Groups. J Fourier Anal Appl 26, 26 (2020). https://doi.org/10.1007/s00041-020-09739-5
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DOI: https://doi.org/10.1007/s00041-020-09739-5