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"nonelliptic problems" (subelliptic Lap!acians, degenerate elliptic operators, etc.), but remain entirely within the pseudodifferential framework in the sense.
Internat. J. Math. & Math. Scl. (1985)75-91

75

Vol 8 No.

FINITE PROPAGATION SPEED AND KERNELS OF STRICTLY ELLIPTIC OPERATORS DAVID GURARIE Department of Mathematics Case Western Reserve Lniversity Clee!and, Ohlo 44106 (Received September 20,

ABSTRACT. and discuss

|984)

We establish estimates of the resolvent and other related kernels LP-theory for a class of strictly elliptic operators on in

The clas of operators considered in the paper is of the form

A0

leading elliptic part

LP-type

have

and a "singular" perturbation

and are modeled after

Schrdinger

B,

A0 + B wlth he whose coefficients

operators.

KEY WORDS AND PHRASES. Elliptic operator, resol’env, semigro.p, pert,a-5on series, finive propagavion speed. 192 MATHEMATICS SUBJECT CLASSIFICATION CODES. 47G05; 35S05 I.

INqRODUCTION.

_

In this paper we shall study the resolven= kernel and other related "functions of

AO [ aa

(x)D

a

on

n

A"

,

(-A)

-I

for a class of strictly elliptic op=rators

and their perturbations

We are =articu!ar]v_. interesEed in p

R_(x,y)

LP-theory

A o

0

+ B. elliptic operators,

and typical probies :hau arise here are spectral properties ef

A,

c!esedness, essential se!fadcintness, accre:ivity, semigroup generation, etc.

Underlying ali those kernel

;s

ehe qu=ion of existence and estimates of the reeivent

R_.

Typically the estimate ha3 :he form of a convolution-t::pe bound

xth an L

i-r adial

decreasin functlcn H In the uniformly elliptic case kernels of the resolvent R_

semiroup

K

e

-tA

(-A)

were studied extensively ([!], [2], [3] eta!.).

particular, Eidelman [i] derived :he following radial bound of

K

-i

In

and

D. GURARIE

76

where

order

m

A,

m

m

and I. Gelfand and G. Shilov [2] applied this

m-I’

A.

estimate to study generalized eigenfunction expansions of

In [4], [5] we obtained similar estimates in a different way starting with The latter approach allows us to treat perturbations of

the resolvent kernel.

and consequently operators with "nonregular" coefficients.

LP-spectral

results to

A,

We applied these

theory of elliptic operators.

In the present paper we extend the results of [4] to a class of strictly elliptic operators with possibly unbounded coefficients, obtained by linear deformations of uniformly elliptic symbols in the -variable, a

(X,x())

a(x,)

depending smoothly on

Here x E

x

n.

x

means a matrix function

(ij(x))

Natural examples of such deformations are second order elliptic operators

aij(x)D2...

A

Indeed any quadratic form 2

as a deformation of the simplest one:

Il

a(x,) 6

()I 2 x

aij(x)i with

can be viewed

" (aij (x))=.

x Other examples appear as right (left) invariant operators on nilpotent Lie groups

(see [6]). For the sake of presentation we shall restrict ourselves to the simplest case of deformations when

x

(x)l

is scalar

("conformal dilations").

transformations correspond to a multiplication of the leading symbol of

(x)

a positive function

m,

m

order of

A.

The dilating factor

subject to a certain constraint, called "finite propagation

Its precise definition is given in growth of

at

coefficients of

,

A

2.

(x)

Such

A with is

speed" condition.

This condition limits the rate of

which can not exceed

can not grow faster than

O(Ixl)

xl

m

In other words leading for m-th order operators.

Let us observe that growth restrictions on the leading coefficients are well known in both ordinary (Sturm-Liouville) theory,

A--

d

2

+ -a(x).--dx

(see for instance [7], Ch. 9) and also for partial

differential operators

A

A

I

aij

(x)D’2"lj +

([8]). A sufficient condition for Weyl’s "limit circle case") is

to be well defined (essentially selfadjoint or

the divergence of the integral

+

dx

+ on

(x)

dr

(r)

(a(r)

(x)!l) Ixl

(x)

is estimated by the usual

K(x,y) < bound (I.A).




< R;

and large

,

i

in any regioD

lard I >

}

L

and

perturbation

d


}. the shape of QR,"

F

One such example consists of two rays

Angle

can be

88

D. GURARIE

e

A

An operator

THEOREM 5. -tA

of Theorem 3 generates a holomorphic semigroup p min p Re t > 0 in all L I




(p

"finitely propagating"

Ll-radially

are no longer sufficient for

O(Ixl)

be "slower" than

Namely, an

i)

Ll-case

.

bounded.

K

and a

the growth of

and some additional relation between

Ixi

I

becomes more subtle

bounded

LI:

and a

L

K

6

must

appears.

and

is K(lK(x,y) It is well known (see, for instance, [17], Appendix) homogeneous of degree p y-PK(l-y) dy < =. The latter is obviously that K is L if and only if I. but fails for p true for p > i, as 2vrx on R and write the operator XAMPLE 2. Now we take (x)

EXAMPLE I.

(x) -I.

Take

on

Then

J

K6f

I

(K6f)(x)

(x)y) dy.

K(y)f(x

Then

f K(y)I

IIKflll

We introduce a new variable whole real line

R

and on the interval line

and

Rx 13

_y2

0

u

(d) 0