2
STEVE HOFMANN AND JOHN L. LEWIS
ft = {X = (XQ,X) : XQ ^(x), x G
Rn
*} , where tp is a Lipschitz function on
Rn~1 (i.e. |T/(X) ~'lP(y)\ c|x 2/|, for some positive c, whenever x,y G it!71-1). Let
p(^o, x) = (#o + ^(^)
5
#) # £ i ?
n _ 1
.
Then clearly p maps C/ = {(^o, x) : x0 0, x G jR n - 1 } onto Q and dU onto 9Q in a
one to one way. If u is a solution to Laplace's equation in O, then it is easily seen
that u = ii o p satisfies weakly in U a pde of the form
(1.2) V (AVu) = 0
where A = A(x) is symmetric, satisfies standard ellipticity conditions, and has
coefficients independent of XQ (depending only on x). From this fact one can see
that at least in spirit the pde involving A, can be differentiated with respect to XQ
to get that uXQ also satisfies this pde. Using this idea and a Rellich identity, Jerison
and Kenig were able to show that the Radon Nikodym derivative of harmonic
measure (defined relative to (1.2) and with respect to some point in 17) is in a
certain L 2 reverse Holder class with respect to Lebesgue measure on dU, whenever
A is symmetric, satisfies standard ellipticity conditions and is independent of x$.
Next we consider the analogue of this result for the heat equation in a time
varying graph domain of the type considered by Lewis-Murray[LM] and Hofmann-
Lewis[HL]. To this end suppose that ip = 'ip(x,t) : Rn~l x R^R has compact
support and satisfies
(1.3)
\ip(x, t) ip(y, t)\ ai\x y\, for some a\ oo, and all x, y G Rn~1,t G R.
Also let D | /
2
^ ( x , t ) denote the 1/2 derivative in t of ij;(x,-),x fixed. This half
derivative in time can be defined by way of the Fourier transform or by
JR
%«*.0.c/_«'^-ff-"
for properly chosen c. Assume that this half derivative exists for a.e (x, t) G Rn and
D[/2^ G B M O ( i T ) with norm,
(1.4) 11^1/2^11* «2 OC.
Here BMO(Rn) (parabolic BMO) is defined as follows: Let Q = Qd(x, t) = {(y, s) G
Rn : \yi Xi\ d, 1 i n 1, |s t]1/2 d} be a rectangle in Rn and given
/ : Rn-^R1 locally integrable with respect to Lebesgue n measure, let
fQ = \Q\~1 [ f(x,t)dxdt
JQ
where dxdt denotes integration with respect to Lebesgue n measure and \E\ denotes
the Lebesgue measure of the measurable set E. Then / GBMO(i? n ) with norm \\f\\*
if and only if
s u p d Q r 1 / \f-fQ\dx} oo.
Q JQ
We note that (1.3), (1.4) imply and are only slightly stronger than (1.3) and
(1.5)
\tp{x,t) -ip(x,s)\ c(ai + a2) \s - t\1/2 for some c 0 and all x £ i ? n _ 1 , s , i R
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