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)