Banach Spaces of Analytic Functions by J. Baker, C. Cleaver, J. Diestel, G. Bennett, S.Y. Chang,

By J. Baker, C. Cleaver, J. Diestel, G. Bennett, S.Y. Chang, D.E. Marshall, J.A. Cima, W. Davis, W.J. Davis, W.B. Johnson, J.B. Garnett, J. Johnson, J. Wolfe, H.E. Lacey, D.R. Lewis, A.L. Matheson, P. Orno, J.W. Roberts

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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)).

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