Estimate of the \(L^p\) -Fourier Transform Norm for Some Lie Groups

This chapter aims to survey some recent results on the estimate of the norm of the \(L^p\) -Fourier transform ( \(1 < p \leq 2\) ) of some Lie groups. We focus on the setting of connected nilpotent Lie groups, the semidirect product \(\mathbb {R}^n\rtimes K\) (and also some universal covering groups), where \(K \) stands for a compact subgroup of automorphisms of \(\mathbb {R}^n\) , and the case of compact extensions of locally compact groups. We highlight on the fact that the orbit method and the Plancherel theory play a crucial role in getting the results of the estimates, without any hope to get extremal functions, except in some particular cases.