3. MULTIPLICATIV E VS . ADDITIV E FORMULATIO N 3
The Weierstrass nowhere differentiable function
sin(7r2/cx) E-
2k
k=0
is a typica l elemen t o f th e Zygmun d clas s A* . Not e tha t i f (j) i s differentiabl e a t
the poin t x, the n th e lef t han d sid e o f (1.6) vanishe s a s \h\ 0. Thi s lead s t o th e
so-called Zygmund A * class.
D E F I N I T I ON 1.3. A real valued function (j) G A* is in A * if
U m
\j{x + h) + l(x-h)-2t(x)\
= Q
|/iHo \h\
uniformly in x.
Functions i n thi s clas s ma y b e quit e irregular , fo r instanc e
^ c o s ( 2 f c x )
fc=l
is i n A * and i s almos t everywher e no t differentiabl e (se e [78 , pg . 47]) .
2. Characterizatio n (2)
a
: Introducin g B M O an d V M O
We wan t t o characteriz e C 1,a domains . T o thi s en d w e sa y tha t Q i s C 1,a i f
and onl y i f it s outwar d uni t norma l n i s i n C a. I t i s know n tha t a vecto r value d
function h : R - R n i s i n C a (R ) i f an d onl y i f
(1.8) su p ^-^ [\h-hi\dxC,
r0, \I\=r r a \I\ Ji
where / denot e intervals , |/ | thei r length , an d hi denote s th e averag e o f h o n /
(see [29 ] an d reference s therein) . I f w e le t a = 0 i n th e abov e condition , w e obtai n
the spac e o f function s o f bounde d mea n oscillatio n (BMO) o f Joh n an d Nirenber g
[38]. Howeve r suc h definitio n ma y no t b e use d t o measur e th e regularit y o f th e
unit norma l sinc e \n\ = 1. Henc e w e introduc e th e so-calle d VMO (Vanishing
Mean Oscillation) class , wher e h G VMO i f an d onl y i f h G BMO an d
" -
ml
r^O, \I\=r \I\ Ji
h hi\dx = 0 .
The clas s VMO play s th e sam e rol e vi s a vi s BMO tha t continuou s function s
play wit h respec t t o th e L°° space .
3. Multiplicativ e vs . additiv e formulation : Introducin g th e doublin g
condition
Conditions (l)
ai
an d (2)
a
ar e i n a certai n sens e "additive" . W e wil l als o nee d
corresponding "multiplicative conditions" (fitte d t o Bore l measures) , se e fo r in -
stance [42 , Pg . 77] . Fo r example : I f (j) : R - R , an d (j)' exists, the n (1.6) an d (1.3)
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