4 1. ANALYTIC APPROAC H
L E M M A 1.3. For every x,y G D with x j ^ y, let us set
rj(x,y) = VygD{x,y).Vp\y)
and let T(x, y) be defined by
^V9D(X, y) = ri(x, y)Vp\y) + T(x, y).
There exists a constant C(D) such that, for every x,y G D, x ^ y?
\V(x,y)\ C(D)p(x)\x -
y\~d
and \T{x,y)\ C(D)p(x)p(y)\x ~
y\-^d.
We recall that the Poisson kernel of D is the continuous function
kD : DxdD (0, oo)
defined by
i f)
kD(x,y) =
~2"^r(x'2/)' x G D'y e dD-
Assume that / is a bounded or nonnegative Borel function on dD and define for
every X G D ,
KDU)(X)=
/ kD(x,y)f(y)dy.
JdD
The operator KB is called the Poisson operator of D.
If Kp(f) is bounded, it is harmonic in D. If, in addition, / is continous on dD,
then Ku(f) is the unique solution of the Dirichlet problem
JAu = 0
in
D
\u\dD = /•
Bounds for the Poisson kernel The Poisson kernel satisfies the following global
upper and lower bounds.
L E M M A 1.4 (see [MaPl], Lemma 6 and [CZ], Section 5.2). There exists a
constant C(D) 0 such that for every (x, y) G D x dD,
C(D)-lp(x)\x
-
y\~d
^ kD(x, y) C(D)p(x)\x -
y\~d.
1.2.1. Poisso n integral representation. We denote by H = H(D) the set
of all nonnegative harmonic functions in D and we denote by J\f = N(D) the set
of all finite Radon measures on dD. For every v G A/", we define
hv{x) = KD{v)(x) : = / kD{x1y)v{dy), x G D.
JdD
This definition extends the Poisson operator defined above. In particular, KJJ(V) G
H for every v G M. It is well-known that the mapping
KD : M{D) —+ W(D)
v i /i„
establishes a one-to-one correspondence between M and H.
It is possible t o describe analytically the inverse mapping of KJJ. If h = hu =
KD(V), then, for every continuous function (p on dD,
v,(p=\im h{y + rNy)p(y)dy.
ri°
JdD
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