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

TL;DR

The paper demonstrates a fundamental failure of elliptic regularity estimates for the $ar{ ext{d}}$-operator in the context of $J$-holomorphic curves when using $C^k$-norms instead of Sobolev norms, impacting regularity theory.

Contribution

It provides a simple counterexample showing the breakdown of elliptic bootstrapping for $J$-holomorphic curves in $C^k$-norms, highlighting limitations in regularity methods.

Findings

Calderón-Zygmund estimate fails in $C^k$-norms

Elliptic bootstrapping does not hold in this setting

Implications for regularity theory of $J$-holomorphic curves

Abstract

We present a simple example for the failure of the Calder\'on-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.