INTRODUCTION

Xll l

Fine topologies on the boundary. In 1996, Kuznetsov observed that the Euclid-

ian topology is not appropriate for the definition of the trace and that it should be

replaced by a finer topology (see [DK98c], Section 4). Such a topology is defined

by Dynkin in [Dy98]. For this fine topology which we denote by r, a Borel subset

r of dD is closed if and only if SG(iir) C T. However, proving that SG(tt) is closed

for r for every u £ U has remained an open problem. For this reason, Dynkin and

Kuznetsov introduce in [DK98b] in the general setting of equation (2) a second

fine topology on the boundary which we denote by r . The definition of r relies on

cr-moderate solutions. For every Borel set T C dD, set

ur = suplujy ; v £ A/i, i/(dD\T) = 0, uv ^ u).

In [DK98b], it is proved that ur ^ ur and that ur is cr-moderate. (If T is compact,

ur is actually the greatest r-moderate solution dominated by ur)- By definition, a

Borel set T C dD is closed for f if and only if SG(2

r

) C T (see Section 2.4.2).

The definition of the fine trace. In [DK98b], the fine trace of a solution u of

(3) is defined as the pair

tr(u) = ( I »

where

r =

SG(u)

and v is the cr-finite measure on dD characterized by

v(B) = sup{/i(£) ; lie Mi, /i(r) = 0, u^ ^ u}

for every Borel set B C dD. (See [DK98b] and Section 2.4.3). Note that this

definition is given in [DK98b] in the general context of equation (2). It coincides

with the rough trace in the subcritical case. If r , r ; C dD and v, v' G Mo, we say

that (r , v) and (r ; , v') are equivalent and write (T, v) ~ (r' , v') if and only if v — v'

and TAT' is boundary polar.

If u G U and %T(U) = (T, z/), Dynkin and Kuznetsov prove in [DK98b] that

(A) r C dD is closed for the topology f (i.e. SG(2

r

) C T).

(B) v is a cr-finite measure of class Mo such that u(dD\T) = 0 and S G ^ ) C I \

Conversely, if a pair (I\ v) satisfies (A) and (B), then

t r ( ^ r S ^ ) ^ (r,i/).

Moreover, u? 0 uv is the minimal solution with a trace equivalent to (r , v) and the

only one that is a-moderate.

Key problems. In the survey [DK98c], Section 7, Dynkin and Kuznetsov present

the key remaining problems (see also [Dy02]. The first one can be stated as follows:

P R O B L E M 1. Is every solution a-moderate ?

By the results presented above, a positive answer to this problem would com-

pletely solve the problem of the classification of the solutions of (3). The following

intermediate question is suggested by Dynkin and Kuznetsov.

P R O B L E M 2. Is ur cr-moderate for every Borel subset T of dD ?

In fact, this is equivalent to saying that u-p — UY for every Borel set Y C dD. A

positive answer to this problem would in turn imply that the fine topologies r and

r coincide. Note that it has already been proved that ^ r is cr-moderate, provided

T is relatively open in dD ([Dy02], Theorem 8.1).