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• W2035276572 abstract "Let <italic>D</italic> be a bounded domain in the complex plane. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{H^infty }(D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Banach algebra of bounded analytic functions on <italic>D</italic>. The corona problem asks whether <italic>D</italic> is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=weak Superscript asterisk> <mml:semantics> <mml:msup> <mml:mtext>weak</mml:mtext> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>text {weak}^ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> dense in the space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German upper M left-parenthesis upper D right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>M</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {M}(D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of maximal ideals of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{H^infty }(D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Carleson [3] proved that the open unit disc <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Delta 0> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{Delta _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is dense in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German upper M left-parenthesis normal upper Delta 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>M</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {M}({Delta _0})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Stout [9] extended Carleson’s result to finitely connected domains. Behrens [2] found a class of infinitely connected domains for which the corona problem has an affirmative answer. In this paper we will use Behrens’ idea to extend the results to more general domains. See [11] for further extensions and applications of these techniques." @default.
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• W2035276572 title "A class of infinitely connected domains and the corona" @default.
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