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.

中文翻译：

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$$ L ^ p $$ Lp-关于局部紧凑组的紧凑扩展的傅立叶变换范数

令G为I型可分离的单模局部紧致群，令N为I型G的单模封闭正态子群，使得G / N为紧。设\（1 <p \ le 2 \），\（\ Vert {\ mathscr {F}} ^ p（G）\ Vert \）和\（\ Vert {\ mathscr {F}} ^ p（N） \ Vert \）表示相应的\（L ^ p \）- Fourier变换的范数。我们显示\（\ Vert {\ mathscr {F}} ^ p（G）\ Vert \ le \ Vert {\ mathscr {F}} ^ p（N）\ Vert \）。在特殊情况下，\（G = K <imes N \）由可分离的单模局部压缩群N的半直接乘积定义I型和紧凑子组的ķ自同构组的Ñ，我们表明，等号成立当Ñ具有ķ -invariant序列\（（\ varphi _j）_j \）的函数\（L ^ 1（N）\上限L ^ p（N）\）使得\（{\ Vert {\ mathscr {F}} \ varphi _j \ Vert _q} / {\ Vert \ varphi _j \ Vert _p} \）趋于\（\ Vert {当j变为无穷大时，\ mathscr {F}} ^ p（N）\ Vert \）。我们进一步证明，在某些情况下，N的极值函数扩展到G的极值函数。