By Joseph L. Doob
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Extra resources for Classical Potential Theory and Its Probabilistic Counterpart (Classics in Mathematics)
2) yields u(O) when b -' +co. ) In particular, it follows that a bounded analytic function on the plane is identically constant (Liouville's theorem). Application to Local Properties of Families of Harmonic Functions Harnack's inequality implies that a family u, of positive harmonic functions on a connected open set D is locally uniformly bounded if bounded at a single point. 3) with a near 0 shows that the family u, + I and therefore also the family u. is equicontinuous at each point and thus uniformly continuous on each compact subset of D.
B) In (a) if B = B(g, S) and if u is harmonic on B. so that the restriction to In fact, under the stated conditions, B of u - r,u is positive superharmonic, vanishes at , and therefore vanishes identically. (c) In the definition of superharmonicity in Section 4, (c) can be replaced by (c'). We have already seen that superharmonic functions satisfy (c'). Conversely, if u satisfies Section 4(a), (b), (c') the reasoning leading to the superharmonic function minimum theorem and thereby to (a) above remains 6.
AM,,j, that is, replace u by the corresponding vector lattice maximum (u - ah) Y (ah - u) furnished by Theorem 14. To finish the proof of the theorem, we prove that under (Rl)-(R3) the function u/h has radial limit 0 at C. Let a (9) and a,,(6) be, respectively, the M. and M,, measure of the closed spherical cap on aB cut off by the closed cone of revolution with vertex the origin, axis the radius to , half-angle 0 > 0, and define a (0) = a,,(0) = 0. The functions a and a,, are monotone increasing on [0, n] and right continuous except perhaps at 0, and a,(6) > 0 when 0 > 0 in view of (R3).