We consider differential equations of the form Lu = L 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 denotes the differential operator
LQ = (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 C t . O ' J e ^ u H + || G ( t , c J ) ' ) d ( e c , ) u ) / d t | | + || H ( t , f » )LQeK \\ c | | J(t,|» )e*Lu| | ,
where the norms are in L (I;H), p: I -* • 3R is an increasing weight function
and F, G, H, J are suitable functions of t and of j) f = d(j)/dt. Such inequali-
ties 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
2 . . .
obtain L -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.
1980 Mathematics Subject Classification: Primary 35B40; secondary 3^C11,
Key words and phrases: Second order differential operators,Hardy type ine-
Library of Congress Cataloging-in-Publication Data
Amrein, Werner O.
Hardy type inequalities for abstract differential operators.
(Memoirs of the American Mathematical Society,
ISSN 0065-9266; no. 375 (Nov. 1987))
"Volume 70, number 375 (third of 6 numbers)."
1. Differential equations, Partial-Asymptotic theory.
2. Differential operators. 3. Inequalities (Mathematics)
I. Boutet de Monvel-Berthier, Anne, 1948— . II. Georgescu,
V. (Vladimir), 1947- . III. Title. IV. Series: Memoirs of
the American Mathematical Society; no. 375.
QA3.A57 no. 375 510s 87-25476