Stein-Tomas for compact hypersurfaces with nonzero curvature

Finite graph cover. The local reduction of [F1] and the nondegeneracy of the curvature via [F2] produce, at every point, a curved graph chart; the argument of the finite-graph-cover lemma (compactness plus local construction) selects finitely many such charts covering $S$. The compact localization construction of [F1] uses the local graph normals; their sign-independent nonvanishing curvature in [F2] supplies the curved charts without needing a global normal field. Write the resulting pieces as $S_1,\dots ,S_m$ with graphing functions $h_j$ satisfying $\det D^2{h}_j\mathrm{\ne }0$ and localizations ${\mu }_j={\chi }_j\sigma$ as in [F3].

Endpoint convolution bound. Each patch may be rotated to graph coordinates; its translation contributes a modulation of the data. These operations preserve all relevant norms, so [F4] applies in the original coordinates. Since $\check{\sigma }={\sum }_j{\check{\mu }}_j$ and $f\ast \check{\sigma }={\sum }_{j}f\ast {\check{\mu }}_j$, the patch estimates [F4] and the triangle inequality give $\mathrm{\parallel }f\ast \check{\sigma }{\mathrm{\parallel }}_{p_0^{\mathrm{\prime }}}\le ({\sum }_j{C}_j)\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}=:C_S\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}$ for every Schwartz $f$.

The restriction bound. By the $T{T}^{\ast }$ identity [F5] and step 2.1, $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{L^2(\sigma )}^2=⟨f\ast \check{\sigma },f⟩\le \mathrm{\parallel }f\ast \check{\sigma }{\mathrm{\parallel }}_{p_0^{\mathrm{\prime }}}\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}\le C_S\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}^2$, hence $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{L^2(\sigma )}\le C_S^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}$ for every $f\in \mathcal{S}({\mathbb{R}}^n)$.

The range and the extension. By [F6] the restriction bound of step 3.1 gives a unique bounded extension $R:L^{p_0}\to L^2(\sigma )$ and the dual extension estimate $\mathrm{\parallel }Eg{\mathrm{\parallel }}_{p_0^{\mathrm{\prime }}}\le C_S^{1\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2(\sigma )}$; moreover $\mathrm{\mid }Eg\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_{L^1(\sigma )}\le \sigma (S{)}^{1\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2(\sigma )}$, so $E:L^2(\sigma )\to L^{\mathrm{\infty }}$ is bounded. For finite $q\ge q_0$, integrating $\mathrm{\mid }Eg{\mathrm{\mid }}^q\le \mathrm{\parallel }Eg{\mathrm{\parallel }}_{\mathrm{\infty }}^{q-q_0}\mathrm{\mid }Eg{\mathrm{\mid }}^{q_0}$ gives $E:L^2(\sigma )\to L^q$ for every $q\ge q_0$, and dualizing as in [F6] extends $R_0$ uniquely to a bounded $R:L^p\to L^2(\sigma )$ for every $1<p\le p_0$. At $p=1$, $\mathrm{\parallel }{R}_{0}f{\mathrm{\parallel }}_2\le \sigma (S{)}^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_1$ directly, and smooth density with completeness gives the unique extension. Hölder extends the integral adjoint pairing to $L^1$ as well. Thus the full asserted range holds.

Conclusion. Steps 1.1–2.1 sum the endpoint patch estimates into a convolution bound for the full compact hypersurface, step 3.1 converts it into the endpoint restriction estimate, and step 4.1 gives the full range and the adjoint formulation. The constant depends on $S$ through the finitely many patch constants, the geometry of the cover, and $\sigma (S)$.