On a problem of restriction of Fourier transform on a hypersurface (Q2287056)

The paper deals with the restriction problem in the case of a particular type of hypersurface for which the standard theory does not apply. Given a smooth hypersurface \(S\subset \mathbb{R}^{n+1}\) and a nonnegative function \(\psi \) smooth and compactly supported on \(S\), we ask whether is possible to obtain an estimate of the form: \[ \int_{S}\left\vert \widehat{f}(\xi )\right\vert ^{2}\psi (\xi )dS(\xi )\leq C\left\Vert f\right\Vert _{L^{p}\left( \mathbb{R}^{n+1}\right) }^{2}\text{,for any Schwartz function }f\text{,} \tag{1} \] where \(1\leq p\leq \infty \) and \(C>0\) is a constant depending only on \(S\), \(\psi \) and \(p\). In this direction an old result of \textit{A. Greenleaf} [Indiana Univ. Math. J. 30, 519--537 (1981; Zbl 0517.42029)] ensures that if (\(d\mu :=\psi dS\)) \[ \left\vert \widehat{d\mu }(\xi )\right\vert \leq C^{\prime }\left(1+\left\vert \xi \right\vert \right) ^{-r}\text{,}\tag{2} \] for some \(r,C^{\prime }>0\), then (1) is true for \(p=2\left( r+1\right) /\left( r+2\right) \). Typically, conditions like (2) are established by analysing the curvatures of the smooth hypersurface \(S\) (see [loc. cit.]). It turns out that there exist hypersurfaces for which such analysis is inconvenient and Greenleaf's result does not directly apply. The purpose of the present paper is to treat a particular type of hypersurface that falls in the above category, namely, it is considered the surface \(S=S_{\phi }\) given by the graph of the function \(\phi :\mathbb{R}^{n}\rightarrow \mathbb{R}\) defined by \[ \phi \left( \xi _{1},\dots,\xi _{n}\right) =\xi _{1}^{m_{1}}\cdot \dots\cdot \xi_{n}^{m_{n}}\text{,} \] where \(m_{1},\dots,m_{n}\) are nonnegative integers such that \(m_{1}+\dots+m_{n}\geq 2\). More precisely, the author establishes that for \( S=S_{\phi }\) as above and \(p\) such that \(p^{\prime }\geq 2\left(1+\max_{1\leq j\leq n}m_{j}\right) \) (where \(p^{\prime }:=p/(p-1)\)) the inequality (1) holds. The proof uses some methods from [\textit{I. A. Ikromov} and \textit{D. Müller}, Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra. Princeton, NJ: Princeton University Press (2016; Zbl 1352.42001)], the central role being played by the Littlewood-Paley decomposition. Also, the Greenleaf's result is used at some point in the proof.