The Lp-Fourier Transform Norm on Compact Extensions of Locally Compact Groups.

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 ≤ 2 , ‖ F p (G) ‖ and ‖ F p (N) ‖ denote the norms of the corresponding L p -Fourier transforms. We show that ‖ F p (G) ‖ ≤ ‖ F p (N) ‖ . In the particular case where G = K ⋉ 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 (φ j) j of functions in L 1 (N) ∩ L p (N) such that ‖ F φ j ‖ q / ‖ φ j ‖ p tends to ‖ F p (N) ‖ when j goes to infinity. We show further that in some cases, an extremal function of N extends to an extremal function of G. [ABSTRACT FROM AUTHOR]