Results & Lemmas (8)
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Theorem 1
Theorem 1 and also treat the case where E contains points of U as well as dUΌ NOTATION. If S is a plane set then S denotes its closure and…
Theorem 1 and also treat the case where E contains points of U as well as dUΌ NOTATION. If S is a plane set then S denotes its closure and dS its boundary. A(U) denotes the algebra of all continuous functions on U, analytic on U; H°°(U) denotes the algebra of all bounded analytic functions on U; HE{ U) denotes the algebra of all bounded continuous functions on UljE which are analytic on U. If ye U, a representing measure for y with respect to A(U) is a positive borel measure μ on U such that f(y
THEOREM 1.
THEOREM 1. Suppose C is connected. Let F he a subset of dU with zero harmonic measure for each point of U (with respect to U). Then F is a…
THEOREM 1. Suppose C\U is connected. Let F he a subset of dU with zero harmonic measure for each point of U (with respect to U). Then F is a peak interpolation set for H~{U). The proof follows from the following lemma. LEMMA.. With U and F as in the theorem, let X be a compact 33
Theorem 1
Theorem 1 follows from the lemma in exactly the same way as
Theorem 1 follows from the lemma in exactly the same way as
Theorem 1
Theorem 1 follows from Lemma 2 in [2]. (For an alternative approach see the proof of Theorem 4.3 of [3]). We observe that if A(U) is…
Theorem 1 follows from Lemma 2 in [2]. (For an alternative approach see the proof of Theorem 4.3 of [3]). We observe that if A(U) is pointwise boundedly dense in iϋΓ°°(U) then using Theorem 2.1 of [5] we can modify the function / in the lemma so that it is in Hpum\f}(U). Then we can prove that F is a peak interpolation set for HpU(du\F}(U). In the general situation (where C\U need not be connected) the same method yields the following result. If y e U we denote by My the set of all (positive) re
THEOREM 2.
THEOREM 2. Let yell and F S dU. Suppose there is a decreasing sequence Vn of open sets containing F, such that μ(Vn) —* 0 uniformly for…
THEOREM 2. Let yell and F S dU. Suppose there is a decreasing sequence {Vn} of open sets containing F, such that μ(Vn) —* 0 uniformly for μeMy. Then F is a peak interpolation set for Hp(U).
Theorem 2
Theorem 2 holds, for if / is as in the definition of peak interpolation set, with V chosen so that yίV, and g = 1, then we can choose a…
Theorem 2 holds, for if / is as in the definition of peak interpolation set, with V chosen so that yίV, and g = 1, then we can choose a neighborhood W of F so that |1 - f\ < ε on Z7Π W; by Theorem 5.1 of [1] we can approximate / to within ε on compact subsets of W by a sequence {/„} in A(U) with | | / J | ^ 1, so that μ(W) is small for all μeMy. The question naturally arises: suppose μ{F) — 0 for all μ 6 My. Must there exist open sets VnS F such that μ(Vn) -~>0 uniformly for μ 6 Myl This is easi
LEMMA 2.
LEMMA 2. Let F be α subset of dU such that for each ze F there exists δ > 0 such that Fz = F Π w: | w — z | ^ <5/2 is α peak interpolation…
LEMMA 2. Let F be α subset of dU such that for each ze F there exists δ > 0 such that Fz = F Π {w: | w — z | ^ <5/2} is α peak interpolation set for H~nJ{βtδ)(UΓ\ Δ{z, δ)), then F is a peak interpolation set for Hψ(U).
THEOREM 3.
THEOREM 3. Let S be a subset of V such that U is locally simply connected at each point of S Π dU. Then S is an interpolation set for…
THEOREM 3. Let S be a subset of V such that U is locally simply connected at each point of S Π dU. Then S is an interpolation set for H~ndu(U) if and only if: (i) Uf)S is an interpolating sequence for H°°(U), (ii) Sf]dU has zero harmonic measure for each point of U, with respect to U.