u(x n , t) - g(t)}nEN, o limsupU(xn) n---+oo :s U(xo). Consequently, U is upper semicontinuous. 1 and Fubini's theorem. 1 (i), (iv). The result is due to Gardiner (1985); the proof given here can be found in Gardiner and Klimek (1986). 6 Let 0 be an open subset of Rm. (i) Let u, v be harmonzc m 0 and v > o. If ¢ : R ----+ R zs convex, then v¢(u/v) zs subharmonzc mO. (ii) Let u E S1i(O), v E 'H(O), and v > 0 m o.

4. Let f E O(fl, fl'), where fl, fl' formula c C n , and let u E C2(fl'). Verify the 17 Complex dzjJerentzation Conclude that 5. If {c a } aEZ;t C e, the series La Ca is said to be absolutely convergent (to a complex number c) if for any bijection 'P: z+ -- Z~ the series 00 L c",(j) j=O is absolutely convergent and c is its sum. We shall use the standard multi-index notation. If Z = (Zl' ... ,zn) E en and a = (al,"" an) is a multi-index, then zO: = if! z~n , where, by definition, 00 = 1. Moreover, if a = (al, ...