On the restriction conjecture
The restriction problem in harmonic analysis reads as follows: for which sets \(S\subset\mathbb R^d\) and which \(p,\) \(1\leq p\leq2,\) can the Fourier transform of an \(L^p(\mathbb R^d)\) function be meaningfully restricted. Even for \(S\) to be the unit sphere \(\mathbb S^{d-1}\) in \(\mathbb R^d\) the problem is not completely solved for \(d>2.\) By this, the study of other spaces where the restriction problem holds true makes sense. It is known that this is the case for radial functions. In the paper under review the authors consider the space of radial function multiplied by spherical harmonics. The restriction estimate is true there for familiar parameters \(1\leq p<2d/(d=1)\) and \(q=(d-1)p/(d+1)(p-1),\) where the Fourier transform from \(L^p(\mathbb R^d)\) is restricted to \(L^q(\mathbb S^{d-1}).\) Since the publication of this paper new results appeared in this area. We mention, for example, a nice survey paper by \textit{T. Tao} [''Some recent progress on the restriction conjecture'', Fourier analysis and convexity. Boston, MA: Birkhäuser. (2004; Zbl 1083.42008)]; and \textit{D. Oberlin}, [Can. Math. Bull. 48, No. 2, 260--266 (2005; Zbl 1083.42007)].
scientific article; zbMATH DE number 2114462