Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves.
Roughly, one takes a solution $ u $ of a certain PDE (a la Cauchy-Riemann) and tries to find a tranformation $ \Phi $ and a holomorphic $ \sigma $ with $ u = \Phi \sigma$.


Now, the construction starts as follows: it is obvious that, given a family of almost complex structures $ J \in W^{1,p}(B, R^{2n \times 2n}), J^2 = -Id$, where $B \subset \mathbb{C}$ is a small ball around the origin, one can find a trivializing transformation $\Psi \in W^{1,p}(B, Hom\_{\mathbb{R}}(\mathbb{C}^n, \mathbb{R}^{2n})) $, so that $\Psi^{-1}(z) J(z) \Psi(z) = J\_0$ (here $J\_0$ is the standard complex structure).


Is this step really straightforward?


It seems, this should be true by finding a conjugation $\Psi$ at $z = 0$ and then using the standard Implicit Function Theorem, provided that $J \in C^1$. However, here the regularity is $W^{1,p}$ (in particular, $C^0$ by Morrey's theorem).


Geometrically, I understand that the almost complex structure on the trivial bundle $R^{2n} \times B \rightarrow B$ is transformed to the canonical constant $J\_0$.


Thanks in advance (and sorry, if the question is indeed trivial).



A trivialization of an almost complex structure
It seems one can use the fact that $GL(2n, \mathbb{R})$ acts transitively on the space of almost complex structure $C$ by conjugation. One gets a map $GL(2n, \mathbb{R}) \rightarrow C$ of maximal rank. Thus, one can find locally a smooth section $C \rightarrow GL(2n, \mathbb{R}) $. Precomposing this section with $J(z)$ gives the needed canonization.


