Why bootstrapping for $J$-holomorphic curves fails in $C^k$

We present a simple example for the failure of the Calderón-Zygmund estimate for the $\bar{\partial}$-operator when the Sobolev $(k,p)$-norms are replaced by the $C^k$-norms. This example is discussed in the context of elliptic bootstrapping, Fredholm theory, and the regularity of $J$-holomorphic curves.

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