A Frobenius–Nirenberg theorem with parameter

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander–Nirenberg theorem with parameter. The first extends the Newlander–Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster’s proof for the Newlander–Nirenberg theorem. The second concerns a version of Nirenberg’s complex Frobenius theorem and its proof yields a result with a mild loss of regularity.