PARABOLIC OPERATOR S WITH SINGULAR DRIF T TERM S
3
(see also [HL, section 8] for the equivalence of (1.1) and (1.4) to another condition).
Let Q = {(xo,x,t) : Xo ip(x,t), (x,t) G Rn} and suppose that u is a solution to
the heat equation in Q, (i.e, ut = Au). Let
p(x0, x, t) = (x0 + ip(x, t),x, £), (x, t) G Rn,
when (xo,x,t) G U = {(yo,y,s) : 2/o 0 (y? 5) £ i^ n }. Again it is easily checked
that p maps U onto Q and 9/7 onto 9 ^ in a 1-1 way. In this case u uo p satisfies
weakly an equation of the form (1.1) where
BVu{X,t) = il)t(x,t)uXQ{X,t\ (X,t)eU,
and A = A(x,t) is independent of XQ as well as satisfies standard ellipticity con-
ditions. Unfortunately though, rpt(x,t) may not exist anywhere (see the remark
before (1.5)). To overcome this difficulty we consider as in [HL] a transformation
originally due to Dahlberg - Kenig - Stein. To this end let a = ( 1 , . . . , 1, 2) be an
n dimensional multi-index so that if z = (#,£), then
Xaz = {Xx,\H)
A - z E E ( f , X ) .
Let P(z) G Cg°(Qi(0,0)) and set
P\(z) = A- ( n + 1 ) P(A- Q z).
In addition choose P(z) to be an even non-negative function, with f
n
P(z) dz = 1.
Next let P\i/j be the convolution operator
Px^(z) = I Px(z-v)^j{v)dv.
and put
(1.6) p(x0l x, t) = (x0 + P1Xoip{x, t), x, t), when (z0, x, t) G U.
From properties of parabolic approximate identities and (1.3), (1.5), it is easily
checked that lim Pjyorip(y^s) = ip(x,t). Thus p extends continuously to
(yo,y,s)—(x,i)
dU. Also if 7 is small enough (depending on ai, a^), it is easily shown that p maps
U onto Q and dU onto dft in a one to one way. Next observe that if u is a solution to
the heat equation in Q1 then u = u o p is a weak solution to an equation of the form
(1.1) where A satisfies standard ellipticity estimates. Before proceeding further
we note that parabolic measure on dU, defined with respect to this pullback pde
and a point in U, is absolutely continuous with respect to Lebesgue measure on 9/7,
thanks to [LM, ch.3] (in fact parabolic measure defined with respect to a given point
is an Aoo weight with respect to Lebesgue measure on a certain rectangle). Thus
the pullback pde should be a good model for proving mutual absolute continuity of
parabolic and Lebesgue measure.
In chapters I and II of this memoir we study the remarkable structure of this
pullback pde. In chapter I we establish certain basic estimates for parabolic pde's
with singular drift terms and establish
L2
solvability of the Dirichlet problem for
pde's which are near a constant coefficient pde in a certain Carleson measure sense.
In chapter II we remove the nearness assumption on the Carleson measures con-
sidered in chapter I and thus obtain our first main theorem on absolute continuity
of parabolic measure and the corresponding Lq Dirichlet problem. As a corollary
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