The celebrated Newlander-Nirenberg theorem states that on a smooth manifold, an almost complex structure $J$ is a complex structure if and only if it is integrable, namely, the Nijenhuis tensor $N_J$ vanishes. It was known from Hill and Taylor that if $J$ has Hölder regularity above $C^{1/2}$ then $N_J$ makes sense as a tensor with distributional coefficients. However $N_J$ is undefined for generic $C^{1/2}$ tensor due to the failure of multiplication for $C^{1/2}$ functions and $C^{-1/2}$ distributions.

In the talk, we will explore the integrability condition when $J$ has regularity below $C^{1/2}$. We give a necessary and sufficient condition for $J$ being a complex structure (at least) for $J\in C^{1/3+}$ using Bony's paradifferential calculus. If time permitted, I will also talk about how our method may be related to rough path theory in stochastic analysis and the Gromov's non-embedding problem in algebraic topology.

This is an in progress work joint with Gennady Uraltsev.