C o r r e c t i o n to " E s t i m a t e s and solvability"

Ark. Mat., 43 (2005), 143 144

@ 2005 by Institut Mittag-Leffter. All rights reserved

C o r r e c t i o n to " E s t i m a t e s and solvability"

Nils Dencker

The original article appeared in Ark@ fb'r matematik 37 (1999), 221-243.

In the final typesetting, the end of the proof of T h e o r e m 2.4 disappeared. Thus, the following lines should be inserted before R e m a r k 6.3 on p. 242.

End of the proof of Theorem 2.4. Thus, by L e m m a 6.2 we find t h a t (Re a) ~ >_c~

almost everywhere for some uniformly bounded and real Co E L~ S(h 2, g)). P u t A=a~o~=(a-co) w, this only adds ic~b~EOpS(h,9) to R0, then we obtain t h a t R e A > 0 . W h e n R e a > e > 0 , we m a y replace a with a - c and do the same con- struction to obtain Re A = R e ( a - c o ) ~ >_c. This gives the p r e p a r a t i o n (6.11) of P in the case when h n r + 89 a, b}cS(h, g).

In tile general case, we conjugate with E ~, where

(6.21)

E = e x p

(/0

- I m ( r + 8 9

)

CS(1,9 )

is real valued. We find E - 1 ~ S ( 1 , 9 ) and E ~ ( E - 1 ) ~ - ( E - 1 ) ~ E w ~ I modulo an operator in Op S(h 2,g) with L 2 operator n o r m bounded by CoT when

I~I<-T.

In fact, we find ( E - 1 ) / T E S ( 1 , g ) uniformly when

I~I<T<C.

Thus, E ~" is invertible for small enough T, with an L 2 b o u n d e d inverse. We find t h a t

( D r + i f ~ + r ~ ) E ~ ~ E ~ p

modulo Op S(h, 9), where

r0 -- Re r - 89 a, b} - {f, E } E -1 E S(1,9) satisfies the condition that

Im(ro+89 -0.

144 Nils Dencker: Correction to "Estimates and solvability"

Since E ~ B E ~ ' ~ ( b E 2 ) ~ modulo L 2 bounded operators, we obtain the result with b o = b E 2 C S ( h - l , g ) by applying the estimate (6.17) on E ~ u for small enough T, with P = D t + i f ~ J + r ~ . This completes the proof of Theorem 2.4. []

Received February 2, 2005 Nils Dencker

Department of Mathematics Lund University

Box 118 SE-221 00 Land Sweden

emaih dencker~maths.lth.se