Fourier Restriction and the Stein–Tomas Theorem — Examples

The two representatives coincide almost everywhere. The function ${\mathbf{1}}_{S^{n-1}}$ is Borel measurable by [F2], and $G=\widehat{f}+{\mathbf{1}}_{S^{n-1}}$ is measurable. Since $S^{n-1}$ is Lebesgue null by [F3], $G=\widehat{f}$ almost everywhere. Hence $\int \mathrm{\mid }G{\mathrm{\mid }}^{p^{\mathrm{\prime }}}=\int \mathrm{\mid }\widehat{f}{\mathrm{\mid }}^{p^{\mathrm{\prime }}}<\mathrm{\infty }$ by the Hausdorff–Young membership in [F1], and $G$ belongs to $L^{p^{\mathrm{\prime }}}({\mathbb{R}}^n)$ and represents the same class as $\widehat{f}$, namely $F={\mathcal{F}}_{p}f$: two almost-everywhere equal integrable functions define the same quotient class by [F2].

The restrictions differ at every point of the sphere. By construction $G(\omega )=\widehat{f}(\omega )+1$ for every $\omega \in S^{n-1}$, so the pointwise restrictions satisfy $G{\mathrm{\mid }}_{S^{n-1}}=\widehat{f}{\mathrm{\mid }}_{S^{n-1}}+{\mathbf{1}}_{S^{n-1}}$. The difference is $1$ at every point of the nonempty sphere $S^{n-1}$, so the two restrictions are different functions on $S^{n-1}$.

The refutation. Suppose that the pointwise restriction were well defined by the ambient $L^{p^{\mathrm{\prime }}}$ class, that is, that two representatives of one class always have equal restrictions to $S^{n-1}$. Steps 1.1 and 1.2 exhibit two representatives $\widehat{f}$ and $G$ of the same class $F={\mathcal{F}}_{p}f$ whose restrictions differ everywhere on $S^{n-1}$; this contradicts the supposition. Therefore pointwise restriction is not well defined on $L^{p^{\mathrm{\prime }}}$ classes.

The correct convention. The restriction operator $R_0$ of Fourier restriction and adjoint extension operators is defined on the actual functions $\widehat{f}$ with $f\in \mathcal{S}({\mathbb{R}}^n)$, where the pointwise values exist. If a bound $\mathrm{\parallel }{R}_{0}f{\mathrm{\parallel }}_{L^2(\sigma )}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p}$ holds on Schwartz data, density gives its unique bounded extension $R:L^p({\mathbb{R}}^n)\to L^2(\sigma )$, as in Restriction and extension estimates are dual; no pointwise restriction of a general ambient class is asserted.

The cap integral. With the equator omitted at $\delta =1$ as justified in [F1], the cap condition $1-\omega \cdot e_n\le {\delta }^2$ is equivalent to $\sqrt{1-\mathrm{\mid }y{\mathrm{\mid }}^2}\ge 1-{\delta }^2$, that is $\mathrm{\mid }y{\mathrm{\mid }}^2\le 2{\delta }^2-{\delta }^4$, and the chart density is $(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$; hence $\sigma (C_{\delta })={\int }_{\mathrm{\mid }y{\mathrm{\mid }}^2\le 2{\delta }^2-{\delta }^4}(1-\mathrm{\mid }y{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}dy$, and by [F1] this is bounded between $c_n{\delta }^{n-1}$ and $C_n{\delta }^{n-1}$.

The slab volume. The slab is the box $[-c{\delta }^{-1},c{\delta }^{-1}{]}^{n-1}\times [-c{\delta }^{-2},c{\delta }^{-2}]$, a product of $n-1$ intervals of length $2c{\delta }^{-1}$ and one of length $2c{\delta }^{-2}$; its volume is their product ${\lambda }_n(T_{\delta })=(2c{)}^{n-1}{\delta }^{-(n-1)}\cdot 2c{\delta }^{-2}=(2c{)}^n{\delta }^{-(n+1)}$.

The explicit box concentration. For $x\in T_{\delta }$, $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }\le \sqrt{n-1}c{\delta }^{-1}\le {\delta }^{-1}\mathrm{/}100$, while on the cap $\mathrm{\mid }{\omega }^{\mathrm{\prime }}\mathrm{\mid }\le \sqrt{2}\delta$ and $\mathrm{\mid }{\omega }_n-1\mathrm{\mid }\le {\delta }^2$. Hence $\mathrm{\mid }x\cdot (\omega -e_n)\mathrm{\mid }\le (\sqrt{2}+1)\mathrm{/}100$, since $\mathrm{\mid }{x}_n\mathrm{\mid }\le c{\delta }^{-2}\le {\delta }^{-2}\mathrm{/}100$. In particular $\mathrm{\mid }2\pi x\cdot (\omega -e_n)\mathrm{\mid }<1\mathrm{/}2$. By [F2], $\mathrm{Re}{e}^{2\pi ix\cdot (\omega -e_n)}\ge 1\mathrm{/}2$. Integrating and removing the unit-modulus factor $e^{2\pi i{x}_n}$ gives $\mathrm{\mid }E{\mathbf{1}}_{C_{\delta }}(x)\mathrm{\mid }\ge \mathrm{Re}(e^{-2\pi i{x}_n}E{\mathbf{1}}_{C_{\delta }}(x))\ge \sigma (C_{\delta })\mathrm{/}2$. This proves the required bound for every allowed $c$, independently of the unspecified constant in the concentration lemma.

The cap norm. By [F1], $\mathrm{\parallel }{\mathbf{1}}_{C_{\delta }}{\mathrm{\parallel }}_{L^2(\sigma )}=\sigma (C_{\delta }{)}^{1\mathrm{/}2}$ satisfies $c_n^{1\mathrm{/}2}{\delta }^{(n-1)\mathrm{/}2}\le \mathrm{\parallel }{\mathbf{1}}_{C_{\delta }}{\mathrm{\parallel }}_2\le C_n^{1\mathrm{/}2}{\delta }^{(n-1)\mathrm{/}2}$, so the two quantities $\mathrm{\parallel }{\mathbf{1}}_{C_{\delta }}{\mathrm{\parallel }}_2$ and ${\delta }^{(n-1)\mathrm{/}2}$ are comparable with constants depending only on $n$.

The extension lower bound. By step 1.3 the extension of the cap data satisfies $\mathrm{\mid }E{\mathbf{1}}_{C_{\delta }}(x)\mathrm{\mid }\ge \frac{1}{2}\sigma (C_{\delta })$ for every $x\in T_{\delta }$, so for $1\le q<\mathrm{\infty }$ $\parallel \widehat{{\mathbf{1}}_{C_{\delta }}d\sigma }{\parallel }_q=\mathrm{\parallel }E{\mathbf{1}}_{C_{\delta }}{\mathrm{\parallel }}_q\ge \frac{1}{2}\sigma (C_{\delta }){\lambda }_n(T_{\delta }{)}^{1\mathrm{/}q}\ge \frac{1}{2}{c}_n(2c{)}^{n\mathrm{/}q}{\delta }^{n-1-(n+1)\mathrm{/}q},$ and for $q=\mathrm{\infty }$ the same lower bound reads $\ge \frac{1}{2}{c}_n{\delta }^{n-1}$, which is the limiting value of the displayed exponent.

The power comparison. Comparing the two powers of steps 2.1 and 2.2, the extension estimate with a constant uniform in $\delta$ requires ${\delta }^{(n-1)-(n+1)\mathrm{/}q}\lesssim {\delta }^{(n-1)\mathrm{/}2}$ as $\delta \downarrow 0$, that is $(n-1)\mathrm{/}2\le n-1-(n+1)\mathrm{/}q$, or equivalently $q\ge 2(n+1)\mathrm{/}(n-1)$; this is exactly the necessary condition of [F3] and the conclusion of the Knapp example.

On the coordinate box $T_{\delta }$, $\mathrm{\mid }(x_1,\dots ,x_{n-1})\mathrm{\mid }\le \sqrt{n-1}{c}_n{\delta }^{-1}=a_n{\delta }^{-1}$ and $\mathrm{\mid }{x}_n\mathrm{\mid }\le c_n{\delta }^{-2}\le a_n{\delta }^{-2}$, so this box lies in the cylindrical tube supplied by [F2]. Therefore [F1] and [F2] give $\mathrm{\parallel }E{g}_{\delta }{\mathrm{\parallel }}_q\mathrm{/}\mathrm{\parallel }{g}_{\delta }{\mathrm{\parallel }}_2\ge \frac{1}{2}\sigma (C_{\delta }{)}^{1\mathrm{/}2}{\lambda }_n(T_{\delta }{)}^{1\mathrm{/}q}\ge \frac{1}{2}(c^{\mathrm{\prime }}{)}^{1\mathrm{/}2}(2c_n{)}^{n\mathrm{/}q}{\delta }^{(n-1)\mathrm{/}2-(n+1)\mathrm{/}q}$. This is an inequality with an explicit positive constant, and its exponent $a=(n-1)\mathrm{/}2-(n+1)\mathrm{/}q$ is negative by [F3].

Divergence. Since $a<0$, ${\delta }^a\to +\mathrm{\infty }$ as $\delta \downarrow 0$; hence for every $C>0$ there is $\delta \in (0,1]$ with $\mathrm{\parallel }E{g}_{\delta }{\mathrm{\parallel }}_q>C\mathrm{\parallel }{g}_{\delta }{\mathrm{\parallel }}_2$, which refutes the existence of any finite extension constant $\mathrm{\parallel }E{\mathrm{\parallel }}_{L^2(\sigma )\to L^q}$ for $q$ below the Tomas exponent.

The restriction form. If a restriction estimate at some finite $p>1$ held with constant $R$, then by the duality of restriction and extension estimates the extension estimate would hold at $q_0=p^{\mathrm{\prime }}$ with the same constant; for $p>2(n+1)\mathrm{/}(n+3)$ one has $p^{\mathrm{\prime }}<2(n+1)\mathrm{/}(n-1)$, which step 2.1 rules out. At $p=\mathrm{\infty }$, the Knapp theorem [F3] also rules out the estimate by its compact norm tests. Hence no restriction estimate exists for $p\in (2(n+1)\mathrm{/}(n+3),\mathrm{\infty }]$, as asserted.

Conclusion. Steps 1.1–3.1 exhibit the cap family $g_{\delta }$ whose extension norms exceed any proposed constant below the exponent $2(n+1)\mathrm{/}(n-1)$, and step 3.1 transfers the failure to the restriction side for $p>2(n+1)\mathrm{/}(n+3)$.

The transform of the flat measure. Since $\mu$ is carried by $\{{\omega }_n=0\}\cong [-1,1{]}^{n-1}$ with density $1$, [F1] gives $\check{\mu }(x)={\int }_{[-1,1{]}^{n-1}}{e}^{2\pi i{x}^{\mathrm{\prime }}\cdot {\omega }^{\mathrm{\prime }}}d{\omega }^{\mathrm{\prime }}$: the variable $x_n$ does not appear. By Fubini over the product $[-1,1{]}^{n-1}$ and the one-dimensional evaluation [F2], $\check{\mu }(x)={\prod }_{j<n}{\int }_{-1}^1{e}^{2\pi i{x}_{j}t}dt={\prod }_{j<n}\frac{\sin 2\pi x_j}{\pi x_j},$ with each factor read as its continuous value $2$ at $x_j=0$.

No decay along the normal. Setting $x^{\mathrm{\prime }}=0$ gives $\check{\mu }(0,x_n)={\prod }_{j<n}2=2^{n-1}$ for every $x_n\in \mathbb{R}$, since the product is independent of $x_n$. Along the normal line $x^{\mathrm{\prime }}=0$ the function is the nonzero constant $2^{n-1}$, so for no constant $C$ can $\mathrm{\mid }\check{\mu }(x)\mathrm{\mid }\le C(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-(n-1)\mathrm{/}2}$ hold for all $x$: as $\mathrm{\mid }{x}_n\mathrm{\mid }\to \mathrm{\infty }$ the right-hand side tends to $0$ while the left remains $2^{n-1}$. This refutes the curvature-free statement, and it shows that the hypothesis in [F3] is necessary.

The sinc product is continuous and positive at $x^{\mathrm{\prime }}=0$, so its modulus is bounded below on a tangential ball of positive measure. It is independent of $x_n$; Tonelli on that ball times $\mathbb{R}$ gives $\int \mathrm{\mid }Eg{\mathrm{\mid }}^q=\mathrm{\infty }$ for every $1\le q<\mathrm{\infty }$, while $g\in L^2(\mathrm{\Sigma })$. To test the smooth-localization hypothesis itself, take a nonnegative nonzero $b\in C_c^{\mathrm{\infty }}({\mathbb{R}}^{n-1})$. Then $\int e^{2\pi i{x}^{\mathrm{\prime }}\cdot {\omega }^{\mathrm{\prime }}}b({\omega }^{\mathrm{\prime }})d{\omega }^{\mathrm{\prime }}$ is independent of $x_n$ and equals $\int b>0$ at $x^{\mathrm{\prime }}=0$, so it also fails the decay estimate. It is the restriction of an ambient smooth cutoff times $b$, and hence is an allowed smooth localized measure on the flat graph. The compact-surface conclusion also needs curvature: a sphere can be modified on its upper graph by replacing $\sqrt{R^2-\mathrm{\mid }y{\mathrm{\mid }}^2}$ with $\chi (y)R+(1-\chi (y))\sqrt{R^2-\mathrm{\mid }y{\mathrm{\mid }}^2}$, where $\chi =1$ on a small ball and vanishes outside a larger ball strictly inside $\mathrm{\mid }y\mathrm{\mid }<R$. The resulting compact smooth embedded hypersurface has a flat open patch. A nonzero smooth density supported in that patch gives the same normal-coordinate independence and rules out every finite-$q$ extension estimate.

Conclusion. Steps 1.1–2.2 exhibit the flat localization $d\mu ={\mathbf{1}}_{[-1,1{]}^{n-1}}d{\omega }^{\mathrm{\prime }}$ whose transform does not decay in the normal direction and whose extension fails every finite-$q$ bound; in particular the curvature hypothesis in the curved-patch decay and in the compact-hypersurface corollary cannot be removed.

The endpoint values. Substituting $n=2$ into [F1] gives $p_0=2\cdot 3\mathrm{/}(2+3)=6\mathrm{/}5$ and $q_0=2\cdot 3\mathrm{/}(2-1)=6$.

Conjugacy. $1\mathrm{/}(6\mathrm{/}5)+1\mathrm{/}6=5\mathrm{/}6+1\mathrm{/}6=1$, so $q_0=p_0^{\mathrm{\prime }}$; equivalently $(6\mathrm{/}5{)}^{\mathrm{\prime }}=6$, and the extension bound at $q_0=6$ is the dual form of the restriction bound at $p_0=6\mathrm{/}5$.

The exponent identity. $1\mathrm{/}{p}_0-1\mathrm{/}{q}_0=5\mathrm{/}6-1\mathrm{/}6=4\mathrm{/}6=2\mathrm{/}3$, while $2\mathrm{/}(n+1)=2\mathrm{/}3$ at $n=2$; this is the identity $1\mathrm{/}p-1\mathrm{/}{p}^{\mathrm{\prime }}=2\mathrm{/}(n+1)$ used in the fractional-integration step of the endpoint proof.

Conclusion. On the circle the Stein-Tomas endpoints are $p_0=6\mathrm{/}5$ and $q_0=6$, the two are conjugate, and the fractional-integration identity reads $1\mathrm{/}{p}_0-1\mathrm{/}{q}_0=2\mathrm{/}3$.