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Searching for a Counter-Example

  1. Oct 30, 2006 #1

    I am searching for a counter-example in analysis.
    Here is the problem :

    I want a differentiable (in the sense of [tex]R^2[/tex]) function [tex] f : \mathbb{C} \to \mathbb{C}[/tex] such that

    1) f is square integrable
    2) [tex]\partial_z f [/tex] is square integrable
    3) [tex]\partial_{\bar z} f [/tex] is not square integrable

    I tried, but I didnt find anything... Except the function [tex]f(z) = \bar z [/tex], which
    is such that 2) and 3) are satisfied, but 1) is not...

  2. jcsd
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