The Newlander-Nirenberg Theorem for principal bundles

Let $G$ be an arbitrary (not necessarily isomorphic to a closed subgroup of $\mathrm{GL}(r,\mathbb{C})$) complex Lie group, $U$ a complex manifold and $p:P\to U$ a $\mathcal{C}^\infty$ principal $G$-bundle on $U$. We introduce and study the space $\mathcal{J}^\kappa_P$ of bundle almost complex structures of H{\"o}lder class $\mathcal{C}^\kappa$ on $P$. To any $J\in \mathcal{J}^\kappa_P$ we associate an $\mathrm{Ad}(P)$-valued form $\mathfrak{f}_J$ of type (0,2) on $U$ which should be interpreted as the obstruction to the integrability of $J$. For $\kappa\geq 1$ we have $\mathfrak{f}_J\in\mathcal{C}^{\kappa-1}(U,\bigwedge\hspace{-3.5pt}^{0,2}_{\,\,U}\otimes\mathrm{Ad}(P))$ whereas, for $\kappa\in[0,1)$, $\mathfrak{f}_J$ is a form with distribution coefficients. Let $J\in \mathcal{J}^\kappa_P$ with $\kappa\in (0,+\infty]\setminus\mathbb{N}$. We prove that $J$ admits locally $J$-pseudo-holomorphic sections of class $\mathcal{C}^{\kappa+1}$ if and only if $\mathfrak{f}_J=0$. If this is the case, $J$ defines a holomorphic reduction of the underlying $\mathcal{C}^{\kappa+1}$-bundle of $P$ in the sense of the theory of principal bundles on complex manifolds. The proof is based on classical regularity results for the $\bar\partial$-Neumann operator on compact, strictly pseudo-convex complex manifolds with boundary.The result will be used in forthcoming articles dedicated to moduli spaces of holomorphic bundles (on a compact complex manifold $X$) framed along a real hypersurface $S\subset X$