Hardy type inequalities for abstract differential operators / Werner Amrein, Anne Boutet de Monvel-Berthier and Vladimir Georgescu

Authors' abstract: We consider differential equations of the form Lu≡L 0 u+S(t)du/dt+T(t)u=w, where u,w are functions defined on an interval I=(a,+∞) with values in some Hilbert space H, L 0 denotes the differential operator L 0 =(d/dt)M(t)(d/dt) and M(t), S(t), T(t) are linear operators in H, t∈I, with M(t) positive, symmetric and converging to the identity operator as t→∞. Our principal results are inequalities of the type ∥F(t,φ ' )e φ u∥+∥G(t,φ ' )d(e φ u)/dt∥+∥H(t,φ ' )L 0 e φ u∥≤c∥J(t,φ ' )e φ Lu∥, where the norms are in L 2 (I;H), φ : I→ℝ is an increasing weight function and F, G, H, J are suitable functions of t and of φ '≡dφ/dt. Such inequalities allow one for example to get information about the asymptotic behaviour of a function u:I→H from that of the function w≡Lu, in particular to obtain L 2 -upper bounds for eigenfunctions of the differential operator L. We give applications to ordinary differential operators and to first order perturbations of the Laplacian; general second order elliptic operators will be discussed in a separate publication. (Zentralblatt)

Bibliogr. p. 116-119