Stein-Tomas theorem for a torus and the periodic Schrödinger operator with singular potential (Q2856440)

The Stein-Tomas theorem provides an estimate for the restriction of the Fourier transform of a test function \(f\in \mathcal S(\mathbb R^d)\) to a compact hypersurface in terms of the \(L_p\)-norm of \(f\) (see [\textit{T. H. Wolff}, Lectures on harmonic analysis. Providence, RI: American Mathematical Society (2003; Zbl 1041.42001)]. The author finds its analog for a hypersurface of a \(d\)-dimensional torus \(\mathbb T^d=\mathbb R^d/\Gamma\) where \(\Gamma\) is a lattice. The space \(L_p(\mathbb R^d)\) is replaced with \(\ell_p(\Gamma')\) for the dual lattice \(\Gamma'\). The result implies the absolute continuity of the spectrum of a periodic Schrödinger operator with a \(\delta\)-potential concentrated on a hypersurface of nonzero curvature.