The Bochner–Riesz conjecture implies the restriction conjecture

Statement. Let $n$ be a natural number with $n \ge 2$. If the Bochner–Riesz estimate (H) below holds in dimension $n$, then the restriction estimate (C) below holds in dimension $n$.

This is a one-way implication, (H) $\Rightarrow$ (C), for each fixed $n \ge 2$. There are no other variables or assumptions: the only inputs are the natural number $n$ and the hypothesis $n \ge 2$.

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Setting and conventions

• Space. $\mathbb{R}^n$ carries the Euclidean inner product $\langle x, \xi\rangle = \sum_{i=1}^n x_i \xi_i$ and the Euclidean norm $|x| = \bigl(\sum_i x_i^2\bigr)^{1/2}$. Its measure $dx$ is Lebesgue measure, so the unit cube has volume $1$.
• Test functions. $\mathcal{S}(\mathbb{R}^n)$ is the Schwartz class of complex-valued functions $f$ that are $C^\infty$ and satisfy $\sup_x |x|^k\,\|D^j f(x)\| < \infty$ for all $k, j \ge 0$.
• Fourier transform. For $f : \mathbb{R}^n \to \mathbb{C}$,

$$\widehat{f}(\xi) = \int_{\mathbb{R}^n} e^{-2\pi i \langle x, \xi\rangle}\, f(x)\, dx ,$$

and the inverse transform of $g : \mathbb{R}^n \to \mathbb{C}$ is

$$\check{g}(x) = \int_{\mathbb{R}^n} e^{2\pi i \langle \xi, x\rangle}\, g(\xi)\, d\xi .$$

Both are Lebesgue (Bochner) integrals, and by convention an integral whose integrand is not absolutely integrable is assigned the value $0$. For Schwartz $f$ the integral defining $\widehat f(\xi)$ converges absolutely, so this convention does not affect $\widehat f$.

• $L^p$ norms. For a measure $\mu$ and an exponent $p \in [1,\infty]$, $\|g\|_{L^p(\mu)}$ is the extended nonnegative number $\bigl(\int |g|^p\, d\mu\bigr)^{1/p} \in [0,\infty]$ when $p < \infty$, and the $\mu$-essential supremum of $|g|$ when $p = \infty$. Every inequality below compares elements of $[0,\infty]$, and the constants $C$ are finite nonnegative reals.

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The hypothesis (H): Bochner–Riesz estimates in dimension $n$

For a real number $\delta$, define the symbol

$$m_\delta(\xi) = (1 - |\xi|^2)_+^{\delta} = \begin{cases} (1 - |\xi|^2)^{\delta} & \text{if } |\xi| < 1,\\ 0 & \text{if } |\xi| \ge 1,\end{cases}$$

where the power is the real power of a positive number. Define the Bochner–Riesz operator of order $\delta$ on functions $f : \mathbb{R}^n \to \mathbb{C}$ by

$$S^\delta f(x) = \int_{\mathbb{R}^n} e^{2\pi i \langle \xi, x\rangle}\, m_\delta(\xi)\, \widehat f(\xi)\, d\xi ,$$

that is, the inverse Fourier transform of $m_\delta \widehat f$ (with the convention above that a non-integrable integrand gives $0$).

For $p \in [1, \infty]$, define the critical index

$$\beta(p) = \max\Bigl( n\,\Bigl|\tfrac1p - \tfrac12\Bigr| - \tfrac12,\ 0 \Bigr),$$

with $\tfrac1\infty = 0$. So $\beta(p) \ge 0$ always, and $\beta(\infty) = \max\bigl(\tfrac{n-1}{2}, 0\bigr) = \tfrac{n-1}{2}$.

(H) asserts: for every exponent $p$ with $1 \le p \le \infty$ (the endpoint $p = \infty$ included) and every real $\varepsilon > 0$, there is a finite constant $C \ge 0$ such that

$$\bigl\| S^{\beta(p) + \varepsilon} f \bigr\|_{L^p(\mathbb{R}^n)} \;\le\; C\, \|f\|_{L^p(\mathbb{R}^n)} \qquad \text{for every } f \in \mathcal{S}(\mathbb{R}^n).$$

The constant $C$ may depend on $n$, $p$ and $\varepsilon$, but not on $f$. The estimate is required only for Schwartz functions, not for all of $L^p$.

Because $\varepsilon > 0$ and $\beta(p) \ge 0$, the order $\delta = \beta(p) + \varepsilon$ is strictly positive. So $m_\delta$ takes values in $[0,1]$ and vanishes outside the open unit ball, and $m_\delta \widehat f$ is a bounded measurable function supported in the closed unit ball. Its inverse-transform integral therefore converges absolutely, and the zero-value convention plays no role in (H).

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The conclusion (C): restriction estimates for the unit sphere in dimension $n$

Let $S^{n-1} = \{\omega \in \mathbb{R}^n : |\omega| = 1\}$ be the unit sphere, regarded as a measure space in its own right. Its measurable sets are the traces on $S^{n-1}$ of Borel sets of $\mathbb{R}^n$. Its measure $\sigma$ is the cone measure

$$\sigma(A) = n \cdot \bigl|\{\, t\omega : 0 < t < 1,\ \omega \in A \,\}\bigr| \qquad (A \subseteq S^{n-1} \text{ measurable}),$$

where $|\cdot|$ is Lebesgue measure on $\mathbb{R}^n$: $n$ times the volume of the open cone over $A$ inside the unit ball, with the origin removed. Its total mass is $\sigma(S^{n-1}) = n \cdot |B(0,1)|$.

For $f : \mathbb{R}^n \to \mathbb{C}$, write $\widehat f|_{S^{n-1}}$ for the function $\omega \mapsto \widehat f(\omega)$ on the sphere. Write $\overline{B}_R = \{x : |x| \le R\}$ for the closed ball of radius $R$ about the origin, and $\operatorname{supp} f$ for the closure of $\{x : f(x) \ne 0\}$.

(C) asserts: for every exponent $p$ with

$$1 \le p < \frac{2n}{n+1}$$

(real division, so the bound is the real number $2n/(n+1)$; this forces $p$ to be finite, and for $n \ge 2$ the range is nonempty because $2n/(n+1) \ge 4/3$), there is a finite constant $C \ge 0$ such that for every real $R \ge 1$ and every $f \in \mathcal{S}(\mathbb{R}^n)$ with $\operatorname{supp} f \subseteq \overline{B}_R$,

$$\bigl\| \widehat f|_{S^{n-1}} \bigr\|_{L^p(S^{n-1},\, \sigma)} \;\le\; C \cdot R^{0} \cdot \|f\|_{L^p(\overline{B}_R,\, dx)} .$$

Remarks on reading (C) literally:

• The factor $R^0$ is the real power $R^0 = 1$, so the bound is just $C\,\|f\|_{L^p(\overline B_R)}$. The constant $C$ depends on $n$ and $p$ only. It is chosen before $R$ and $f$, so it is uniform in the radius $R \ge 1$ and in $f$.
• The same exponent $p$ appears on both sides: an $L^p$ norm of $f$ on the ball controls an $L^p(\sigma)$ norm of $\widehat f$ on the sphere.
• The right-hand norm is taken over the closed ball $\overline B_R$ with Lebesgue measure. Since $f$ vanishes outside $\overline B_R$, it equals $\|f\|_{L^p(\mathbb{R}^n)}$.
• The functions considered are exactly the Schwartz functions with compact support in $\overline B_R$, that is, smooth compactly supported functions. This class is nonempty (it contains $0$ and smooth bump functions), so (C) is not vacuous.