On a Poincare type inequality

Let f be such that its Fourier transform is supported outside the unit ball. We prove the following inequality parallel to \ f \ q(-1) parallel to(Lq') less than or similar to parallel to del(\ f \(q-1))parallel to(Lq'), for q greater than or equal to 2. For q = 2 this is nothing but Poincare inequality, while by using Holder we get parallel to \ f \(q/2) parallel to del(\ f \(q/2))parallel to(L2), which has interesting applications for partial differential equations. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.