# Classical Potential Theory and Its Probabilistic Counterpart by J. L. Doob (auth.)

By J. L. Doob (auth.)

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**Extra resources for Classical Potential Theory and Its Probabilistic Counterpart**

**Example text**

Conversely, if u has this property, let 1J = B(~, b) be a ball with closure in D. It is enough to prove that u(~) ~ L(u,~, b) and even, since u is lower semicontinuous, to prove that u(O ~ L(f,~, b) for every finite-valued continuous function f defined on aB and majorized by u. For such a function f the function v = PI(B,f) is harmonic on B with a continuous extension to ]j obtained by setting v = f on aBo The difference u - v is lower semicontinuous on Ii and positive on aB, so that if G > 0, then u - v ~ - G near aB (for example, on aDo, for Do a slightly smaller concentric ball).

2) for sufficiently small b, depending on ~, the function is superharmonic. 2) reduces to A(u,~, b), and the proof needs no change in the general case. EXAMPLE (The Fundamental Kernel and the Green Function of a Ball). 2) the function G(~,·) is for each point ~ harmonic on [RN and is superharmonic on [RN. 5 that this function is harmonic on [RN - {(}. Moreover this function is continuous on [RN, and in view of the local nature of superharmonicity proved in (d) above we need only observe, to prove that the function is superharmonic on [RN, that the superharmonic function average inequality is trivially satisfied at ~.

N. Hence If N = 2 the corresponding evaluation of IV yields -nf(e) = -n;J(e)/2, and the proof of the theorem is now complete. D Extension. If v is a signed measure on IRN for which the integral defining Gv converges absolutely and if the projection of v on some bounded open set D is determined by a bounded density g relative to IN, then (GV)IDE C(1)(D) because if f=glD and dJi=fdIN , then Theorem 7 is applicable to GJi which differs on D from Gv by a harmonic function. The same argument shows that if g is continuous on D and satisfies a Holder condition there, then (GV)'D( -C(2)(D) and ~Gv = -n~g there.