On the norm of the Lp -Fourier transform on compact extensions of ℝ n

Abstract

Abstract This paper aims to prove that the norm of the Lp -Fourier transform of the semidirect product ℝ n ⋊ K $\mathbb {R}^n\rtimes K$ is A p n , where 1 < p ≤ 2 $1 < p \le 2$ , q = p / ( p - 1 ) $q=p/(p-1)$ , A p = p 1 2 p q - 1 2 q $A_p=p^{\frac{1}{2p}}q^{\frac{-1}{2q}}$ , and K stands for a compact subgroup of automorphisms of ℝ n . An extremal function is given by an extension of a Gaussian function. Besides, as an example of non-compact extension, the universal covering group of the Euclidean motion group of the plane is also treated and an estimate of the norm is obtained.