An \(L^{p}\)-\(L^{q}\)-version of Morgan's theorem for the \(n\)-dimensional Euclidean motion group
The \(n\)-dimensional Euclidean motion group \(M(n), n \geq 2,\) is the semidirect product of \({\mathbb R}^n\) with \(K = SO(n)\). The Haar measure of \(M(n)\) is \(dx\,dk\), where \(dx\) is the Lebesgue measure on \({\mathbb R}^n\) and \(dk\) is the normalized Haar measure on \(K\). Denote by \(\widehat{M}(n)\) the unitary dual of \(M(n)\). A description is given of the infinite-dimensional elements of \(\widehat{M}(n)\) that are sufficient for the Plancherel theorem. Let \(\mu\) be the Plancherel measure on \(\widehat{M}(n)\). The support of \(\mu\) is parametrized by \((r, \lambda) \in ] 0, \infty [ \times \widehat{U}\), where \(U = SO(n-1)\) is the subgroup of \(SO(n)\) leaving fixed \(\varepsilon_n = (0, \dots, 0, 1)\) in \({\mathbb R}^n\). For \(f \in L^1( M(n)) \cap L^2(M(n)), \pi_{r, \lambda} (f)\) is a Hilbert-Schmidt operator on the Hilbert space \(H_{\lambda}\). Moreover, the restriction of \(\mu\) on the part \(] 0, \infty [ \times \{ \lambda \}\) is given, up to a constant depending only on \(n\), by \(r^{n-1} dr\). Set \(\hat{f} (r, \lambda) = \pi_{r, \lambda}(f)\). The main result of the paper is the following: Let \(p, q \in [1, \infty], a,b \in ] 0, \infty [\), and \(\alpha, \beta\) positive real numbers satisfying \(\alpha >2\) and \(1/\alpha + 1/\beta =1.\) Suppose that \(f\) is a measurable function on \(M(n)\) satisfying {\parindent=5mm \begin{itemize}\item[1)] \(e^{a\parallel x \parallel^{\alpha}} f(x,k) \in L^p(M(n)),\) \item[2)] \(e^{br^{\beta}} \parallel \hat{f}(r, \lambda) \parallel_{HS} \in L^q( {\mathbb R}^+, r^{n-1} dr)\) for all fixed \(\lambda\) in \(\widehat{U}\). \end{itemize}} If \((a \alpha)^{1/\alpha} (b \beta)^{1/ \beta} > (\sin(\pi/2)(\beta -1))^{1/\beta}\), then \(f = 0\) a.e. The authors conclude the paper by showing that the above inequality is sharp.
scientific article; zbMATH DE number 5282447