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**Extra info for An Harnak inequality for Liouville-type equations with singular sources**

**Example text**

F i l x = f k keJ 10 k e J '0 -1 Y ~ D . 10 S i n c e ( 3 . 7 ) i s t r u e a n d f T 1 ( b ) + a , w e may a p p l y a l l o u r r e s u l t s -1 t o ( f . 1jeJ 3 fs l u t D . 3 instead ( f j )j e J . 16. REMARK. I n g e n e r a l , Aut D i s n o t T-complete as t h e f o l l o w i n g example shows: L e t u s p u t D=: A a n d d e f i n e f =: Un by means o f Then Un i s T - c o n v e r g e n t t o t h e c o n s t a n t mapping U : 1'1; however U$Aut A . T h i s phenomenon j u s t i f i e s t h e a s s u m p t i o n o f t h e e x i s t e n c e o f a p o i n t asD s u c h t h a t l i m f , ( a ) =bsD i n t h e t h e o r e m .

T h u s , i f were f e A u t D , by t h e o r e m 2 . 2 we would 3 IeJ h a v e T l i m f T 1 = f-l. 7) i s a T-Cauchy sequence. L e t u s w r i t e b . = : f , ( a ) and c h o o s e a n y b a l l s 1 3 B c c D and B ' c c D c e n t e r e d r e s p e c t i v e l y a t a and b s u c h t h a t f ( B ) c B ' . From t h e o r e m 3 . 1 3 i t f o l l o w s t h a t Hence w e c a n f i x 6>0 s o t h a t t h e set C= i s contained i n B. L e t u s i n t r o d u c e a l s o Cf=: 1 W e may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t d D ( b .

T h e n , b y P , u e havE s e t t i n g yp=: xo+p[f (xo) -XI by the Cauchy estimates, we have Proof: If p g n - I , = I( xp-yp/I+ ( / [fP-idD] (xl)-[fP-idD] (x,) :: II Xp-YpIl '1 d-6 II + , Since x 0 = yo and x 1= y 1 by induction d I1 xl-xoII the statement of the lemma follows COROLLARY. 8. 11 fP-idD I1 (fp-idD)-p(f-idD)] I B C. ,p-I.. In pariicular we have w h e n e v e r p~p(&),w h e r e p(6)=: maxtp: 1) fk-idDI/ <6 f o r a22 kcp}. a Proof: We have x -y = [fP(x)-x]-p[f P P (x)-x] whence the assertions are immediately obtained by the arbitra- 51 COMPLETE VECTOR FIELDS r i n e s s of xcB.