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I am interested in how one could prove that $H(\Omega)\subset \mathscr C^1(\Omega)$ without using the Cauchy integral formula, or any other complex analytic techniques, only real analysis.

Thank you in advance

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Why would you want to do that? That said, it's probably possible to show that $\bar\partial$ is elliptic, or at least hypoelliptic using purely real methods. – mrf Jun 14 '13 at 15:20
But I'm afraid this will not be enough : it does not seem obvious that if a function $f$ is holomorphic (i.e. just differentiable in the complex sense) then $\overline\partial f=0$ in the distribution sense. – Etienne Jun 14 '13 at 18:59

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