"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
