F. and M. Riesz theorem for CR functions (Q1775181)

We recall that a CR manifold \(M\) is {minimal} at a point \(x_0\in M\) if there is no CR submanifold \(N\) with \(x_0\in N\subset M\), \(\dim_{\mathbb{R}}N<\dim_{\mathbb{R}}M\), and \(H_xN=H_xM\) for all \(x\in N\). When \(M\subset\mathbb{C}^n\), the embedding of such a submanifold \(N\) constitutes a singular CR measure in \(M\). The authors show that in fact this is the essential obstruction to the regularity of CR measures, by proving that a CR measure on \(M\) is represented by an \(L^1_{\text{loc}}\) density in a neighborhood of any point \(x_0\in M\) where \(M\) is minimal. This also implies that the weak limit, in the sense of the Radon measures on \(M\), of a sequence of restrictions to \(M\) of entire functions in \(\mathbb{C}^n\) is an \(L^1_{\text{loc}}\) function.