Hartogs's extension theorem: Difference between revisions

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:Let {{mvar|f}} be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G'' \ ''K''}}, where {{mvar|G}} is an open subset of {{math|'''C'''^''n''}} ({{math|''n'' ≥ 2}}) and {{mvar|K}} is a compact subset of {{mvar|G}}. If the [[Complement (set theory)|complement]] {{math|''G'' \ ''K''}} is connected, then {{mvar|f}} can be extended to a unique holomorphic function {{mvar|F}} on {{mvar|G}}.{{sfnm|1a1=Hörmander|1y=1990|1loc=Theorem 2.3.2}}

TheEhrenpreis' proof is based on the existence of smooth [[bump function]]s, unique continuation of holomorphic functions, and the [[Poincaré lemma]] — the last in the form that for any smooth and compactly supported differential (0,1)-form {{mvar|ω}} on {{math|'''C'''^''n''}} with {{math|{{overline|∂}}''ω'' {{=}} 0}}, there exists a smooth and compactly supported function {{mvar|η}} on {{math|'''C'''^''n''}} with {{math|{{overline|∂}}''η'' {{=}} ''ω''}}. The crucial assumption {{math|''n'' ≥ 2}} is required for the validity of this Poincaré lemma; if {{math|''n'' {{=}} 1}} then it is generally impossible for {{mvar|η}} to be compactly supported.{{sfnm|1a1=Hörmander|1y=1990|1p=30}}

The ansatz for {{mvar|F}} is {{math|''φ f'' − ''v''}} for smooth functions {{mvar|φ}} and {{mvar|v}} on {{mvar|G}}; such an expression is meaningful provided that {{mvar|φ}} is identically equal to zero where {{mvar|f}} is undefined (namely on {{mvar|K}}). Furthermore, given any holomorphic function on {{mvar|G}} which is equal to {{mvar|f}} on ''some'' open set, unique continuation (based on connectedness of {{math|''G'' \ ''K''}}) shows that it is equal to {{mvar|f}} on ''all'' of {{math|''G'' \ ''K''}}.