2 1. MOTIVATIO N AN D STATEMEN T O F TH E MAI N RESULT S
an unbounde d domain . I f a functio n v G C°°(fl) satisfie s
{
Av = 0 i n tt
v 0 i n Q ,
v\an = 0 ,
then w e cal l i t minimal positive harmonic function i n Q. W e denot e b y duo 00 th e
harmonic measure with pole at infinity,
d"°° = J^(Q)da(Q).
dnQ
Here th e functio n v play s th e sam e rol e a s th e Poisso n kerne l fo r a bounde d
domain. A s a n example , i f Q = R™
+1
= {(x, t) G
Mn + 1
| t 0} , then v(x, t) = t an d
du°° = d x o n M n. A s w e sho w below , th e classe s o f domain s tha t ar e studie d i n thi s
book ar e s o genera l tha t th e classica l surfac e measur e ma y no t b e wel l defined . I n
that cas e w e substitute da wit h th e restrictio n t o dfl o f the n-dimensiona l Hausdorf f
measure i n R n + 1.
Two basi c (an d related ) question s w e inten d t o addres s are :
Question 1. What is the relationship between the regularity of the domain and the
doubling character of its harmonic measure?
Question 2 . What is the relationship between the regularity of the domain and the
smoothness of its Poisson kernel?
In particular , th e result s w e ar e goin g t o describ e originate d fro m tryin g t o
understand i n th e asymptoti c limi t a s a 0 , th e followin g well-know n results :
T H E O R E M 1.1 (Kellog g [40]) . If Q c W 1 is of class C 1 ^, 0 a 1, then
duj kda, and log/ c G C a,
and it s converse ,
T H E O R E M 1.2 (Al t an d Caffarell i [2]) . IfilcM 71 satisfies certain (necessary)
"weak conditions" (to be specified later) and logk G Ca, then ft is of class C 1,a.
1. Characterizatio n (l)
a
* Approximatio n wit h plane s
The firs t ste p i n answerin g Question s 1 an d 2 consist s i n finding th e correc t
formulation o f th e result s above , a s a 0. T o d o this , w e nee d t o recal l tw o
real-variable characterization s o f th e Holde r classe s C a an d C l,0i:
Let 0 a 1 an d j b e real-valued . W e sa y tha t (j G C
1,a
(IR
n
) i f an d onl y i f
there exist s C 0 suc h tha t fo r an y r 0 an d xo G Mn ther e i s a n affin e functio n
Lr^Xo o n R n satisfyin g
(i.5) w*)-w*) i
CrQ) fo r | x
_
Xoi r
r
At a = 0 , w e hav e (l)o , whic h ha s th e equivalen t formulatio n (Zygmund's A *
class)
(1.6) C , fo r ever y x G it.
\h\
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