4

STEVE HOFMANN AND JOHN L. LEWIS

we obtain the results of [LM] mentioned above. In chapter III we obtain parabolic

analogues for pde's with singular drift terms of theorems in [FKP].

We emphasize that our results are not straight forward generalizations of theo-

rems for elliptic equations. For example we do not know if the pde's we consider

in chapter 2 have parabolic measures which are doubling, as is well known for the

corresponding elliptic measures. Also we cannot prove certain basic estimates such

as Holder continuity for the adjoint Green's function of our pde's. In this respect

our work is more akin to results of [VV] and [KKPT] in two dimensions. Finally

we mention that the possible lack of doubling for our parabolic measures forces us

in chapter III to give alternative arguments in place of the usual square function -

nontangential maximum arguments.

The first author would like to thank Carlos Kenig for helpful discussions con-

cerning necessary conditions on A, B to prove Theorem 2.13. The second author

would like to thank Russell Brown and Wei Hu for useful discussions concerning

basic estimates for pde's with drift terms.

2. STATEMENT OF RESULTS

As rationale for the structure assumptions on our pde's, we shall briefly outline

the structure of the pullback pde under the mapping given in (1.6). To this end

recall that a positive measure // is said to be a Carleson measure on U if for some

positive c oc

/i[(0,d) x Qd(x,t)] c\Qd(x,t)\ for all d 0, (x,t) eRn.

The inflmum over all c for which the above inequality holds is called the Carleson

norm of /x and denoted ||/x||. The following lemma is proved in [HL, Lemma 2.8].

Lemma A. Let o, 0 be nonnegative integers and f = (0i,... , 0

n

_i), a multi-index,

with I = a-\-\(p\Jr0. Ifi/) satisfies (1.3), (1-4) for some ai, a2 oo, then the measure

v defined at (xo,x,t) by

d

_ ( dlpyXQi \ 2 (2J+2A-3)

dxdtdxn

is a Carleson measure whenever either a + 0 1 or |0| 2, with

i/[(0,d) xQ

d

(x,t)] c\Qd(x,t)\.

Moreover, if I 1, then at (xo,x,£)

I

d l p

^ I J (fl1 .

a

\ ri-i-e

where c' = c'(n) and c = c(ai, a2,7, Z,n) 1.

Remark. The last inequality in Lemma A remains true under the weaker assump-

tions (1.3), (1.5). We shall use this remark in chapters II and III.

Recall that in section 1 we introduced the pullback function, u — uop, where p is

as in (1.6). Also u satisfied a certain pullback pde of the form (1.1). We note that

a typical term in the pullback drag term B Viz, evaluated at (A, t), is -^P1Xoxjj uXQ.

From Lemma A with cr = 0 = |/|, 0 = 1, we see that

dfi{X,t) = x0[§-tP1XQ^(x,t)}2dXdt