8 STEVE HOFMANN AND JOHN L. LEWIS
for all (x,t)
eRn,d
0. Here c* = c*(e0,7i, M, A,n).
Remark. 1) In Theorem 2.13, x$B denotes the 1 by n matrix function,
(AT, t) xoB(X11). We note that the smallness assumption in Theorem 2.13 can be
weakened. We do not prove this weakened version since its proof is more compli-
cated and since we are primarily interested in the case when ||/ii|| -f ||/i2|| + \\nz\\
is large. We refer the reader to the remark at the end of section 5 for an exact
statement of a stronger form of Theorem 2.13.
2) To prove the above result for small eo 0, we shall use local estimates in [A],
[M], [Ml], and [FGS] for solutions to the pde in (1.1) and its adjoint pde, but we
will also need to show that if a solution to (1.1) or its adjoint pde has continuous
zero boundary value on Q2d{x-)t) C dU, then this solution is Holder continuous on
(0, d) x Qd(x, t). The proof of Holder continuity cannot be deduced from the usual
arguments (such as reflection) since the drag term B evaluated at (AT, t) can blow
up as xo^O (almost like
x^1)
even under the above Carleson measure assumptions
on B. Using these basic estimates it is not difficult to show that the continuous
Dirichlet problem for the pde in (1.1) always has a unique solution. Moreover,
we can use these estimates to modify slightly an argument of Fabes and Safonov
[FS] to show first that the adjoint Green's function corresponding to (1.1) satis-
fies a backward Harnack inequality in U when eo is sufficiently small and second
that parabolic measure corresponding to (1.1) is a doubling measure. To be more
precise, we show that
v(d, x,t + 2d2, Q2r{y, s)) cu(d, x,t + 2d2,Qr(y, s))
whenever (x,t) G Rn,d 0, and Q2r{y,s) C Qd{x,t) provided eo is sufficiently
small. We shall make our basic estimates and prove doubling for pde's of the form
(1.1) in section 3.
In section 4 we begin the proof of Theorem 2.13. We first show that parabolic
measure is in the above reverse Holder class when £? = 0 and all the above Carleson
norms are small. In this case we perturb our results off a constant coefficient pde
by making estimates of the form:
(5||7V(|VG|)||L
2 ( 0 2 x o
(,,t)) ||S(/0|Ua(1»),
where 5—0 as eo—0 while G is the Green's function for (1.1) with B = 0 and pole
at (#o, x,
£ + 2XQ).
Also h is a weak solution to V-
(.AQV/I)
= 0 while N, S are defined
below in (2.14), (2.15). We note that if
y-1\(A-A0)oo\2(Y,s)dYds
were a Carleson measure on [7, then the above estimate would be an easy con-
sequence of Cauchy's inequality and (2.16) at the end of this section. Since this
measure need not be Carleson (compare with the prototype equation discussed in
section 1), we are forced to integrate by parts numerous times in xo,x,t and use all
of our Carleson measure assumptions on A, in order to obtain the above estimate.
The case when B 0 and eo is small, follows easily from the above case using our
basic estimates and another perturbation type argument. The proof of Theorem
2.13 is given in sections 4 and 5.
/ (A - A0)00 Gyohyo dYds
KQx
0
W)
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