4
1. MOTIVATIO N AN D STATEMEN T O F TH E MAI N RESULT S
can b e formulate d i n the followin g wa y
(1.9)
(1.10)
G A* i f and onl y if
G A* i f and onl y if
\Ir\Jr/ \Ii\l
h
\IAJI/~W\JL
c.
o(l),
\Il\JI,
where I
r
= [x,x + h], and / / = [x h,x\. I n order to state the corresponding "multi-
plicative conditions" w e define positive Borel measures duo = e^ dx an d assum e fo r
a moment that , wit h respec t t o the weight // , the operations o f exponentiation an d
averaging ca n b e commuted , tha t \s/f j exp ft « exp/f
7
j) f (here /fj denote s averag e
over / an d « denote s a two-side d boun d wit h multiplicativ e constants) . Thi s i s
obviously no t tru e i n general, bu t fo r instanc e i t hold s if exp0' G A00(dx) (se e [25 ]
for mor e details) . Give n thi s assumption , the n (1.9), an d (1.10 ) ca n b e rephrase d
as follows :
(l.n)
(A* class) Fo r an y x i n R, on e ha s
C ,
(1.12) (A * class) Fo r an y x i n R , on e ha s l + o(l) , a s |ft | -0 .
Conditions (1.11 ) and (1.12) give a rough notion of "regularity " fo r the measur e
du. I n fact , (1.11i ) s equivalent t o duo bein g a doubling measure (se e below fo r th e
precise definition) , whil e (1.12 ) i s i n som e sens e a n optimal doubling condition .
However, th e reade r shoul d kee p i n min d tha t i n genera l suc h measure s coul d b e
purely singula r wit h respec t t o th e Lebesgu e measur e dx o n R , se e [5 ] an d [9] .
The "multiplicative" analogue , i n term s o f a; , o f conditio n (2) o i s give n b y duo =
kdx an d lo g A: G VMO (se e M. Korey [49 ] an d [50]) .
4. Characterizatio n (l)
a
an d flatnes s
Next, w e introduce a geometri c versio n o f (l)o , namel y th e notio n o f "Locally
Bat domains". Thi s wil l allo w u s t o stat e som e geometri c measur e theor y result s
which hav e (l) o a s a point o f departure .
We begin b y recalling th e definitio n o f Hausdorff distance D betwee n tw o sub-
sets A, B o f R
n + 1
(se e als o [24]) : W e say tha t D[A, B] S if an d onl y i f A i s in a
S—neighborhood o f B an d B i s in a S~ neighborhood o f A, i.e .
(1.13) D[A, B] = ma x ( sup{d(a , B)\a G A}; sup{d(A, b)\b G B} J .
DEFINITION
1.4. We say that Q C
Rn + 1
is S-Reifenberg flat if and only if for
every compact set K C R
n + 1,
there exists RK 0 such that if Q G K n dQ and
0 r RK, then there exists an n—dimensional plane L(r,Q) containing Q and
such that
(1.14)
1
D[ B(r, Q)nd£l; B(r, Q) n L(r, Q) } S.
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