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).