8 1. PRELIMINARIES
COROLLARY
1.2.4. Let Q, = Q'xl and T = T'xl where
Qf
is a domain in
Rn
,
r7
is a ( nonempty)
C2-hypersurface
in R" which is an open part of dO!
and I is an interval in
R1.
Then the estimate (1.2.4) with m = (2, ... , 2, 1)
is valid.
IfQ = {\x\ 1, xn 0} then the estimate (1.2.3) with p(x) = exp(-crxj
- 1 is valid as well for sufficiently large a .
PROOF. AS
in the proof of Corollary 1.2.3 we may assume
Q7
is the lower
half-ball {\x\ 1, xn 0} in R" . Then using new variables yn+l = xn+l ,
yn = xn + g(x", xn+l), y" = x" , we reduce the general case to the one when
a = {\x\ 1, xn 0} and T = dQ n {xn 0} .
With p(x) = exp(-crxn) - 1 we have
C = (^i tn-\ in ~
iTa
exp(-axj , £„+1).
From the equality Am(x\ C) = 0 we have the first equality (1.2.9) and the
equality
The left-hand side in (1.2.2) is
a2
cxp(-axn)(4{^2
aJn£j)2
+
4(ann)2T2a2
exp(-2axn)) +
where denotes the second sum in the expression for the left-hand side
under consideration in the elliptic case. As above, choosing a large, we get
positivity of this expression. So Corollary 1.2.4 follows from Theorem 1.2.1.
In Corollary 1.2.5 we consider a hyperbolic operator
A{x; D) =
cD2n+l
- A + ] P
aiDj
+ a (the sum over j = 1, ... ,n + 1).
We introduce the function
(1.2.10) cp{x) =
exp((a/2)(x2
+ +
x„2_1
+ (x„ -
b)2
-
dx2n+{
- s))
and denote by QE the subset of Q where
COROLLARY
1.2.5. Let Q = G x ( - 7 \ T) a ^ T = r
/
x ( - 7 \ T) w/zere
G /5 a domain in R" ,
r7
c 9G, am/ 0 T. Assume that either
1) //*£ ongzz belongs to G,
T7
= dG, 5 = 6 = 0, or
2) G c {-/z xn 0}, 0 h, V = dG n {x„ 0}, 0 6, s =
d2
+
b2,
where d = max\x"\ over x e Q.
Suppose that
(1.2.11)
6(c + xn+lDn+lc/2 + \xn+{Vc\c
1/2)
1 + {{x , Vc) - bDnc)/(2c),
0c, 6c\ onQ, Q6,
dist2(dG,
0)
6T2in
case 1), h(h + 2b)
BT1
in case 2).
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