6 STEV E HOFMANN AND JOHN L. LEWIS
Under these conditions and standard ellipticity assumptions, we shall show the
Radon-Nikodym derivative of parabolic measure on rectangles is in a certain reverse
Holder class when /^, 1 i 3, are small.
In order to state the main theorem in chapter I precisely we introduce some
notation which will be used throughout this memoir. For completeness we restate
some of the notation used earlier. Let G, 9G, |G|, denote the closure, boundary, and
Lebesgue n, n 4-1 measure of the set G, whenever G C Rn or Rn+l and there is no
chance of confusion. If G C Rn, let LP(G), 1 p oo, be the space of equivalence
classes of Lebesgue measurable functions f on G which are p th power integrable
with norm denoted by
||/||LP(G)-
If G 1S open let
CQ°(G)
be infinitely differentiable
functions with compact support in G. For k a positive integer let Hk{G) be the
Sobolev space of equivalence classes / whose distributional partial derivatives D@ f
( P = (/?o? A? /?n-i) = niulti - index) of order k are square integrable. Let
\Hk(G)
( E i ^ / i
2
)
1 / 2
\j3\k
\L*(G)
and put H$(U) equal to the closure in Cg°(U)
oiHk{U).
We say that / G H\QC (G),
Lp,
(G), if / G
Hk(G1),Lp(G1),
respectively, whenever Gi is open with G\ C G.
Let Lp{Ti,T2,HK (G)),l p oo, k a positive integer, be equivalence classes
of Lebesgue measurable functions / : G x (Xi,T2)—»i? with /(•,£) G iffci (G) for
almost every t G (Ti,T2) and
ll/(^)ll^
( G l )
^ oo,
X
Ti
whenever G\ is open with Gi C G. Lp(Ti,T2, Lp, (G)) is defined similarly with
#
fc
lo c
(G) replaced by
LP1QC
(G).
n-1
As introduced earlier, V = (#!"••• #^r—)
w n n e
^ '
=
5 J ^~- U
n
l
e s s
oth-
2 = 0
erwise stated c will denote a positive constant depending only on the dimension,
not necessarily the same at each occurence, while c(/3, /i, v) will denote a constant
depending only on /3, /i, v. Also points in i? n + 1 will be denoted by (X, t) or (xo, x, t)
while Qd(%,t) C i?n will denote the rectangle with center (#,£), side length 2d in
the space variables, and side length 2d2 in the time variable. We write
Qd(X, t) = (x0 -d,x0 + d)x Qd(x, t) C Rn+l
when there is no chance of confusion. Let f3p(Qd(x,t)) be the reverse Holder class
of functions / :
Rn^R
with \\f\\Lp(Qd(x,t)) °°
a n
d
(2.6)
IQrfas)]-1
I f*dxdt
\P(\Qr(y,s)\-1
f fdxdty
JQr{y,s) JQr{y,s)
for some A, 0 A oo, and all rectangles with Qr(y, s) C Qd(x, £). Let ||/||/3 (Qd(x,t))
be the infimum of the set of all A such that (2.6) holds. Similarly, let av(Qd(x,t))
be the weak reverse Holder class of functions / defined as above except
(2.7)
[QrfasT1
[
fpdxdt\P{\Q2r(y,s)\-lf
fdxdty
JQAv,s) JQ2r(y,s)
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