Notes on a theorem of Baouendi and Rothschild (Q1898391)

The theorem referred to in the title concerns the vanishing of functions which are harmonic in the upper semi-circle and non-negative on the appropriate segment of the real line and which satisfy certain growth conditions as \(y\downarrow 0\). The paper contains results of the following type: let \(f\in L^\infty (\mathbb{R}^+)\), \[ \int_0^\infty f(x) {\textstyle {1\over y} K ({x \over y})} dx= O(y^m) \qquad \text{as} \quad y\downarrow 0, \quad \forall a>0, \] and \(f(x)\geq 0\) a.e. for \(0< x<a\) then \(f(x) =0\) a.e. for \(0<x <a\). Theorems are proved for various kernels \(K\), in one or more dimensions.