Stein-Tomas spherical restriction theorem

Summing the graph-patch bounds. Let $\sigma ={\sum }_j{\mu }_j$ with ${\mu }_j={\chi }_j\sigma$ the finite chart localization of [F1]. For $f\in \mathcal{S}({\mathbb{R}}^n)$, rotate each patch to graph coordinates; orthogonal changes preserve Lebesgue norms and Schwartz space, and chart translations only modulate the data by a unit character. Thus [F2] applies in ambient coordinates. Now $\check{\sigma }={\sum }_j{\check{\mu }}_j$ and hence $f\ast \check{\sigma }={\sum }_{j}f\ast {\check{\mu }}_j$ as everywhere-defined bounded continuous functions. By [F2] and the triangle inequality for $L^{p_0^{\mathrm{\prime }}}$, $\parallel {\sum }_{j}f\ast {\check{\mu }}_j{\parallel }_{p_0^{\mathrm{\prime }}}\le {\sum }_j\mathrm{\parallel }f\ast {\check{\mu }}_j{\mathrm{\parallel }}_{p_0^{\mathrm{\prime }}}\le \left({\sum }_j{C}_j\right)\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}=:C_n\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}\mathrm{.}$ This is the convolution bound at the endpoint.

The restriction bound at $p_0$. Applying the $T{T}^{\ast }$ identity [F3] to the sum of step 1.1, $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{L^2(\sigma )}^2=⟨f\ast \check{\sigma },f⟩\le \mathrm{\parallel }f\ast \check{\sigma }{\mathrm{\parallel }}_{p_0^{\mathrm{\prime }}}\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}\le C_n\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}^2,$ so $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{L^2(\sigma )}\le C_n^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_{p_0}$, which is (a).

By [F4], the endpoint restriction estimate gives $\mathrm{\parallel }Eg{\mathrm{\parallel }}_{q_0}\le C_n^{1\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_2$. Also $\mathrm{\parallel }Eg{\mathrm{\parallel }}_{\mathrm{\infty }}\le \sigma (S^{n-1}{)}^{1\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_2$ by the definition. Applying [F5] with $\theta =q_0\mathrm{/}q$ gives $\mathrm{\parallel }Eg{\mathrm{\parallel }}_q\le (C_n^{1\mathrm{/}2}{)}^{\theta }\sigma (S^{n-1}{)}^{(1-\theta )\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_2$ for $q_0\le q<\mathrm{\infty }$, and the supremum estimate gives $q=\mathrm{\infty }$. For $1<p\le p_0$, duality gives the unique restriction extensions with the same norms. At $p=1$, $\mathrm{\parallel }{R}_{0}f{\mathrm{\parallel }}_2\le \sigma (S^{n-1}{)}^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_1$ directly; density and completeness give its unique extension. The pairing $\int Eg\bar{f}=\int g\overline{Rf}d\sigma$ extends from Schwartz tests by Hölder, also for $p=1$. Conversely, norm testing in $L^2(\sigma )$ shows that each restriction norm is bounded by its extension norm; testing the extension against $L^1$ functions gives the reverse inequality at $p=1$. Thus the adjoint pairing and equality of norms hold throughout the asserted range.

Sharpness. By [F6], the existence of a restriction estimate at exponent $p$ forces $p\le p_0$, and the existence of an $L^2(\sigma )\to L^q$ extension estimate forces $q\ge q_0$; the Knapp cap family $g={\mathbf{1}}_{C_{\delta }}$ with $\delta \downarrow 0$ exhibits both failures. This proves (c).

Conclusion. Steps 1.1–2.1 prove the endpoint restriction bound (a), step 3.1 gives the full range and the adjoint formulation (b), and step 4.1 records sharpness (c) from the Knapp necessary condition.