## Abstract

This paper aims to prove that
the norm of the *L ^{p}*-Fourier transform of the semidirect product ${\mathbb{R}}^{n}\u22caK$ is

*A*

_{p}

^{n}, where $1<p\le 2$, $q=p/(p-1)$, ${A}_{p}={p}^{\frac{1}{2p}}{q}^{\frac{-1}{2q}}$, 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.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.