An Lp-Lq-Version of Morgan's Theorem for the n-Dimensional Euclidean Motion Group

An aspect of uncertainty principle in real classical analysis asserts that a function f and its Fourier transform ̂ f cannot decrease simultaneously very rapidly at infinity. As illustrations of this, one has Hardy’s theorem [1], Morgan’s theorem [2], and BeurlingHörmander’s theorem [3–5]. These theorems have been generalized to many other situations; see, for example, [6–10]. In 1983, Cowling and Price [11] have proved an Lp-Lq-version of Hardy’s theorem. An Lp-Lq-version of Morgan’s theorem has been also proved by Ben Farah and Mokni [7]. To state the Lp-Lq-versions of Hardy’s and Morgan’s theorems more precisely, we propose the following. Let a,b > 0, p,q ∈ [1,+∞], α≥ 2, and β such that 1/α+1/β = 1. If we consider measurable functions f on R such that