Capacities of compact sets in linear subspaces ofRⁿ

Boundary Interpolation for Continuous Holomorphic Functions

“…and BZ functions are those sets for which Cpp(K) = 0. Here, C x p^ denotes a particular (one dimensional) Bessel capacity defined by means of potentials on T. See [5] and [17] for a discussion of this fact as well as [2] for a discussion of the equivalence of the zero sets of corresponding Bessel and Besov capacities. THEOREM B.…”

“…THEOREM B. (Peller and Khruschev [15] and Sjôdin [17]) Let <B = B p l/p nC(T) have the sum norm. If K is a compact subset ofT then the following conditions are equivalent:…”

“…We are also able to provide new information concerning boundary interpolation for the case n = 1, 1 </?<oo, and 1-/3/7^0. Sjodin, [17], has shown that a necessary condition for K to be a B.I. set for B p^ PI C(T) or H$ Pi C(T) is that Cpp(K) = 0.…”

“…Let \i be such a measure. Denote by <j> the function in C(K) which is identically 1 on K. By (iii) there is a sequence of functions {/*} in the unit ball of Hp n C(S) ( [12] for a way to do this when n-1; the general case presents no difficulty because the key point is that, by (17), v can have no point-masses since uj(t) -0{t n~^) .) Now let F G H p. It follows that/ = D?F G H\ and thus F = D~Pf with/ G H\ and, of course, ||F|| w = ||/|| lj0 .…”