PARABOLIC OPERATORS WITH SINGULAR DRIFT TERMS
5
is a Carleson measure on U. Thus a natural assumption on B is that
dfii(X,t) =
x0\B\2{X,t)dXdt,
is a Carleson measure on U with
(2.1) ||/xi||/?ioo.
Next observe from the above lemma with 0 = \j)\ 0, a 1, that
x^-^pixMx,t)\2
dXdt
is a Carleson measure on U. Unfortunately a typical term in AVu evaluated at
(X,i) is
[-^:Pjxo^]2 ux0j
1 i n 1, and for each such i, the measure with
density
x^UrP^]2
dXdt
need not give rise to a Carleson measure. The failure of this measure to be Carleson
makes the structure of A for the pullback pde complicated and causes us to make
an abundance of assumptions on A (all are needed in the estimates and all are
satisfied by our model term, as can be deduced from Lemma A). First assume that
(2.2) (solV^I +x20\At\)(X,t)Aoo
for a.e (X, t) eU and if
dto(X, t) = (xQ \VA\2 + xl \At\2 )(X, t) dXdt,
then /X2 is a Carleson measure on U with
(2.3) ||M2|| 02 00.
Second assume that whenever 0 z, j n 1, we have
n - l
dx0 ~ l^i \ 1 ' dxth 1 ^ y
1=0
in the distributional sense. Here
ij / ij ij ij \
e l V e /1 e Z2 * ' ' ielm -
fij _ / fij fij fij \
h ~ WZl 5 Jl2- ' ' iHru /
are measurable functions from U-+R711 for 0 I n 1 and ( e 3 , j \ 3 ) denotes
the inner product of these functions as vectors in Rni. Third assume that
n - l
(2-4) [ £ |ef'I+ l//'l](*,*) A ^
1=0
and that e3 has distributional first partials in X whenever 0 i,j n 1. In
(2.4),
|eJJ|, \fl3\
denote the norm of these functions considered as vectors in
Rni.
Let
Ve]3
denote the gradient of
e\3
taken componentwise. Fourth assume that
n—1 n 1
dfi3(X,t) = [ E ( E ^ | V e f
|2
+
so"1 \flj\2)
+ \9v\}(X,t) dXdt
ij=0 1=0
is a Carleson measure on U with
(2.5) 11/4,11 (33 00.
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