6 1. PRELIMINARIES
The expression (1.2.2) for T = 0 is defined as the limit as r goes to 0.
This theorem is an anisotropic version of the results of Hormander [42],
Chapter 8, given by Isakov [59]. From the well-known example of H. Lewy
[93] it follows that such a function cp does not exist for every linear differ-
ential operator even with analytic coefficients. Nevertheless, it is possible to
find this function for many important equations and Cauchy problems. In
Corollary 1.2.2 we consider the simplest case: A{x\ Q - (C{ + K2)/2
(the
Cauchy-Riemann operator).
COROLLARY
1.2.2. Let Q be a bounded domain in
R2
and T
eC2
.
i ) / / r c 9 ( l , Ux c Q u r , then
(1.2.5) lMl(0)(n,)
c(||w||(1)(t2))1-A(||wi|(1)(r))A.
ii) If Tea, U{ c Q , then
(1.2.6)
\u\0(Q{)C(\u\0(Q))l-\\u\0(T))\
Here C and A (0 X 1) depend on Qj.
PROOF,
i) We may assume Q is the conformal image of the lower semi-
circle {x\ + x\ 1 , x2 0} and T corresponds to the part of its bound-
ary lying in {x2 0} for in the general case we can cover Q,{ by a finite
number of such domains and T. A conformal mapping of such a domain
onto this semicircle is of class
C1
(fi U T), so using conformal mappings we
may restrict ourselves to the case when Q, is the semicircle mentioned and
r = 9Qn{jc
2
0} .
We apply Theorem 1.2.1 in case a) with the function p(x) = exp(-x
2
)- 1.
If m = (1,1), C = (fi, £2 - iTexp(-x2)) and A{x\ C) = 0, then ^ +
rexp(-x2) = 0 and £2 = 0, so the left-hand side in (1.2.2) is (1/4) exp(-x2)
0 and (1.2.5) follows.
ii) We may assume dQ,{ is Lipschitz. Then applying Theorem 1.1.2 ii)
with n = 2 , k = 2 , p = 2 , and 2 = 0 we get
l«l0(Q,) ciiHiya,)
CIMI^Q^NI^CQ,),
according to Theorem 1.1.2 iv). Here C are constants depending on £lx and
different in different places. We choose two domains Qn and Q12 satisfying
the conditions of part i) so that Q,{ belongs to the union of these domains
and both Q
n
and Q.n are contained in Q. Now let Q2 be a domain with
smooth boundary and Qly c Q
2
, Q2 c Q. According to part i) we have
HMil^fc^,)
c||M||j;-;i)/2(«2)||M||^)2(r).
Using the Cauchy formula one can conclude that
M4(Q2)
C|M|
0
(Q).
According to the interpolation inequalities of Theorem 1.1.1 ii) we have
l«l,(r)
c|«i;/2(r)|M|i/2(r) c|w|i/2(Q)|«i;/2(r).
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