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\
Previous Page Next Page