ABSTRACT

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| | ,

2

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,

35J15, 47F05.

Key words and phrases: Second order differential operators,Hardy type ine-

qualities.

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)."

Bibliography: p.

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

[QA378] [515.7242]

ISBN 0-8218-2438-4

iv