' * ' * ^ Ddvid W Catlin

This implies that at least one of the inequalities

( 2 14) |Li(Reif,_i ,ддт){г')\^от-^ and

( 2 15) \L,(lm^j_,,ddr){z')\^ÔT-^

IS valid (When; = 1, we replace J^^^- i ^ by ^i ^__i ) We may assume that (2 14) holds If we express (2 14) in terms of p, L\, and L\, then it becomes

From (1 23) we obtain that

Thus by the same method used to prove (1 29) we obtain that there exists a small constant y >0 (independent of z\ z, and Ô) so that

( 2 16) \L,{Rc^j.,,ddr){z)\^ÔT-\ zeQ^sizl

Similarly , when J = 1, then

Now set G{z) = Rq^j_, f,ôdr{z) {от if j = 1, set G{z) = Rq{^, ^^,ôôr{z))) Lemma 2.2. There exists a positive number e>0 such that ifzeQysiz') and if (2 17) T{z,eô) = h

then the Hessian of G^ apphed to L = s^L^ -{- S2L2 at z satisfies

( 2 18) ddGHL,L){z)^c''Ô^T-^4s,\^-C'ô-^\s2\^

Proof By successively applying Proposition 1 3, (2 17) (for e still to be chosen), and Corollary 1 4, we obtain that for p = 2, , w,

( eô \p / eS \i

In particular, if p = / - 1, then if; > 1 it follows that for any e > 0, there exists e so that

( 219 ) \^j.,,eêriz)\uCi^,{z)<eÔT-^^\

or in the case when; = 1, that

\^ik - iSdr { z ) \^Ci . , ( z ) <8ÔT - '^'

From (1 23) we obtain the inequality

( 2 20) \L,L,{Rc{^j.,,ddr)){z)\<ÔT-^-'

( The coresponding statements for; = 1 are always obvious so we will omit them ) Observe that

ddG ( L , , L , ) = L,L,G- äG([Li, LJ)