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. r JdD
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