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 \ r i-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
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