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 ∂ -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.