ABSTRACT
As was seen in Chapter 22, there is an obstruction to the local solvability of the tangential Cauchy–Riemann equations at the top degree. In this chapter, we discuss this obstruction in more detail. The system of tangential Cauchy–Riemann equations at the top degree is no longer an overdetermined system of partial differential equations. For example, if f is a form of bidegree (n, n – 1) on a real hypersurface in ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/in2272.tif"/> , then the equation ∂ ¯ M u = f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/in2273.tif"/> consists of one partial differential equation and one unknown coefficient function. If M and f are real analytic, then by the Cauchy-Kowalevsky theorem, there is a local solution to the tangential Cauchy–Riemann equations. For some time, it was thought that ”C ∝” could replace ”real analytic” in the statement of the Cauchy-Kowalevsky theorem. However, in 1957, Hans Lewy [L2] found a counterexample which we present in Section 23.1. We then show that Hans Lewy's example can be recast in the language of the tangential Cauchy–Riemann complex of the Heisenberg group. In particular, Hans Lewy's example provides an example of the local nonsolvability of the tangential Cauchy–Riemann equations at the top degree. In Section 23.2, we consider a more general real analytic, strictly pseudoconvex hypersurface in ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/in2274.tif"/> and we present Henkin's criterion on a smooth form f for the local solvability of the tangential Cauchy–Riemann equations ∂ ¯ M u = f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/in2275.tif"/> .
