We compute the \(L^p\) -Fourier transform norm \(\Vert \mathscr {F}^p(G)\Vert \) and \(L^p\) -Gabor transform norm for several classes of locally compact groups. Let \(\Vert \mathscr {G}^p(G)\Vert \) be the norm of sesquilinear \(L^p\) - \(L^q\) form given by the Gabor transform on the semi-direct product \(G=K\ltimes N\) , where N is a unimodular, separable locally compact group of type I and K a compact subgroup of the automorphism group of N. We prove sufficient conditions under which the equality \(\Vert \mathscr {G}^p(G)\Vert =\Vert \mathscr {G}^p(N)\Vert \) holds. Explicit values of \(\Vert \mathscr {F}^p(G)\Vert \) and \(\Vert \mathscr {G}^p(G)\Vert \) are obtained for groups of type \(\mathbb {H}_n\times D\) , \(\mathbb {H}_n\times \mathbb {R}^m\) , \((\mathbb {H}_n \times D) \ltimes K\) , \((\mathbb {H}_n \times \mathbb {R}^m) \ltimes K\) , where \(\mathbb {H}_n\) is the Heisenberg group, D is a discrete group of type I and K is a compact group.
