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**Example text**

In this case, we may take as defining function p(z) = {-inf(E8D I( inf(E8D I( - - zl, zl, Z z E D; E cn _ \ D. Then D = {z : p(z) < o}. Moreover, when aD E C2 we have the following (see §2 of [213], and also [61]): (a) there is a neighborhood V of aD such that p E C2(V); (b) Igrad pi = 1/2 in V; (c) if z± E V are points on the normal to aD at z such that Iz+ - zl = Iz- - zl, then (aplazk)(Z±) = (aplazk)(Z) and (aplazk)(Z±) = (aplazk)(Z) for k = 1,2, ... ,n. In this case Pk = 2(aplazk) and P'k = 2(aplazk).

1). 1. t -am , Zm aD '>m 8=1 (8 '>8 . 2) = g( (, z) is the fundamental solution to Laplace's equation (see §1). Proof. 2) is proved analogously. Recall that U((,Z) = i)-1)8-1 :%a ((,z)d([s] Ad(. 8=1 Now aF = a a la af -a Zm aD f(() al''>m U((, z) = - aD a(m (fU) + aD a(m U((, z), 1 1 but (_1)8 1aDa~m (f:%a) d([s] Ad(= (_l)n+m 1aDa~8 (f:%a) d(Ad([m], since d (f :%a) d([s] A d([m] = (_1)8-1 a~8 (f :%a) d( A d([m] + (_l)m+n a~m (f :%a) d([s] A de· CHAPTER 1. THE BOCHNER-MARTINELLI INTEGRAL 34 Consequently r ;;!

If z E aD, then we denote by z+ ED and z- ¢. D points on the normal to aD at z such that Iz+ - zl = Iz- - zI. 6. Let *m,k(Z) = laD a~~~z) d([k] A d(, z ¢. aD, m,k(Z) = laD a~~~z) d([k] A d(, z ¢. aD. 9) and these limits are attained uniformly in z. 4. ~=l (8g/8(l)Pf dO"). Notice that dm is a tangential vector field. 10) the limit being attained uniformly in z. Since the points z± lie on the normal to 8D at z, we can write z± - Z = ±gradp· t/I grad pi with t E R. ;(()Pk(Z). ;(()(Pk(() - Pk(Z)), k=l we have Ib((,z)1 :S w(lwl) -+ 0 as Iwl -+ 0 (where w(lwl) is the modulus of continuity of b((, z)). *