On the Fourier transform of measures carried by submanifolds of finite type (Q1361058)

Let \(M\) be a compact finite-type submanifold of \({\mathbb{R}}^n\) of dimension \(m\), with surface measure \(\sigma\). The atomic Hardy space associated to \(M\) is denoted by \(H^1(M)\). The main theorem of this elegant paper is that, if \(\phi \in H^1(M)\), then \(\int_0^{\infty} | (\psi \sigma)\widehat{\phantom{x}} (t\xi)| t^{-1} dt \leq C \| \psi \| _{H^1}\). The proof involves observing that it suffices to take \(\psi\) with very small support, and then applying a change of variables to change the integral involved in computing \((\psi \sigma)\widehat{\phantom{x}}\) to an integral over a small subset of \({\mathbb{R}}^m\), where the simple exponential \(e^{it \xi \cdot x}\) of the Fourier transform is replaced by an exponential with a nonlinear phase. Oscillatory integral techniques are then employed.