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Constructive Theory of Functions of Several Variables by W. Schempp, K. Zeller

By W. Schempp, K. Zeller

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E. on each E,. Thus 9f (x) > l on E, except possibly on a countable union of sets of Cg,p-capacity zero. 6, f E QE, and thus fm fPdv > Cg,p(E), which proves the proposition. 3. 13. Let I < p < oo. Then CG p(E) = Ca, p(E) for any E C RN. Proof. Both capacities are outer, so it is enough to prove the equality for open sets. 2 it is enough to consider compact sets. Let K be compact. 10). p(K) < Ca,p(K). f IIP < CG p(K) + e for some e > 0. We claim that G * f E ? K. Define f by f(x) = min{ f(x), n} for Ixl < it and f(x) = 0 for Ixt > n, so that Ga * f is continuous, and G.

M+(RN) in the weak* topology. Proof. xo g f (x). f (Y) dv(y) < J M lim infg(x,, y) f (Y) dv(Y) 1-' o0 M /J < liminfJ g(x;, y) f(y)dv(y) = liminfQf(x) x-+X0 1 so Cr f is semicontinuous. 3 A General Theory for LP-Capacities 25 (b) Let µ E M±-(RN) and let {µi)' converge to µ weak'. Let be an increasing sequence of continuous functions with compact support on such that h,,(x) converges to g(x, y) for all x. g. Rudin [367], Chapter 2, Exercise 22). 00 RN (c) The proof is similar to (b). Let it and {µ1 } i ° be as in (b), and let {h,,1 be as in (b) but such that h (x) converges to G f (x) everywhere.

1 is a weak solution of the linear second order partial differential equation -Au +u =0 on the complement of K, and one is lead to classical potential theory (with the Newton kernel l2 replaced by the Bessel kernel G2). For p 2, however, the corresponding equations are nonlinear, and much more difficult to handle. For example, for a = I one obtains -div(VulVulp-2) +ululp-2 =0 . ) However, it turns out that by redefining (a, p)-capacity slightly, one is lead to extremal functions that have a simple representation and give a rich potential theory.

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