[proofplan] **This proof is a declared sketch.** We follow Tao--Vargas, "A bilinear approach to cone multipliers I, II," *Geometric and Functional Analysis* **10** (2000), 185--215 and 216--258, and Stein, *Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals*, Princeton University Press 1993, Chapter IX §5 (Bochner--Riesz reduction to Kakeya). The strategy: (1) decompose the multiplier $(1-|\xi|^2)_+^\lambda$ dyadically into annular shells of thickness $2^{-k}$; (2) at $L^2$, the Plancherel bound gives operator norm $\lesssim 2^{-k\lambda}$ per shell (rigorous); (3) at the Kakeya endpoint $p_{\mathrm{Kak}} = 2n/(n-1)$, the Kakeya maximal hypothesis combined with the wave-packet decomposition gives a cited bound for the per-shell operator norm with a critical-exponent factor $2^{k\lambda(p_{\mathrm{Kak}})}\,2^{k\varepsilon}$; (4) Riesz--Thorin between $L^2$ and $L^{p_{\mathrm{Kak}}}$ gives the bound at intermediate $p \in [2, p_{\mathrm{Kak}}]$; (5) the geometric series in $k$ converges precisely when $\lambda > \lambda(p)$. The Kakeya-side bound at $p_{\mathrm{Kak}}$ is the cited content of Stein 1993 Ch IX §5; we state it precisely and use it to close. [/proofplan]

[step:Decompose the multiplier into dyadic annular pieces] Fix a smooth cutoff $\chi: [0, \infty) \to [0, 1]$ supported in $[1/2, 2]$ with $\sum_{k \in \mathbb{Z}} \chi(2^k t) = 1$ for $t \in (0, 1)$. Decompose the Bochner--Riesz multiplier $m_\lambda(\xi) := (1 - |\xi|^2)_+^\lambda$ as \begin{align*} m_\lambda(\xi) = \sum_{k=0}^\infty m_k(\xi), \qquad m_k(\xi) := \chi(2^k (1 - |\xi|^2))\, m_\lambda(\xi). \end{align*} Each $m_k: \mathbb{R}^n \to \mathbb{R}$ is supported in the annular shell \begin{align*} A_k := \bigl\{\xi \in \mathbb{R}^n : 1 - 2^{-k+1} \le |\xi|^2 \le 1 - 2^{-k-1}\bigr\}, \end{align*} of thickness $\asymp 2^{-k}$ in the radial direction, and on this shell $m_\lambda(\xi) \asymp 2^{-k\lambda}$. Hence $\|m_k\|_\infty \asymp 2^{-k\lambda}$. Define the multiplier operator \begin{align*} T_k : \mathcal{S}(\mathbb{R}^n) &\to \mathcal{S}'(\mathbb{R}^n), \\ f &\mapsto \mathcal{F}^{-1}(m_k\, \widehat{f}). \end{align*} Then $T^\lambda f = \sum_{k=0}^\infty T_k f$, with the convergence to be justified once each $T_k$ is bounded with $L^p \to L^p$ norm summable in $k$. [/step]

[step:Bound $T_k$ at the $L^2$ endpoint by Plancherel] By [Plancherel's theorem](/theorems/3179) and the bound $\|m_k\|_\infty \asymp 2^{-k\lambda}$: \begin{align*} \|T_k f\|_{L^2(\mathbb{R}^n)}^2 = \int_{\mathbb{R}^n} |m_k(\xi)|^2\, |\widehat{f}(\xi)|^2\, d\mathcal{L}^n(\xi) \le \|m_k\|_\infty^2\, \|\widehat{f}\|_{L^2}^2 = \|m_k\|_\infty^2\, \|f\|_{L^2}^2. \end{align*} (The last equality uses Plancherel.) Hence \begin{align*} \|T_k\|_{L^2 \to L^2} \le \|m_k\|_\infty \le C\, 2^{-k\lambda}. \end{align*} [/step]

[step:Cite the Kakeya-endpoint bound at $p_{\mathrm{Kak}} = 2n/(n-1)$] We state the per-shell Kakeya-endpoint bound as a cited fact. Set $\delta_k := 2^{-k/2}$, so that the shell $A_k$ has radial thickness $\delta_k^2 = 2^{-k}$ and tangential extent $\asymp 1$. The shell $A_k$ is tiled by smooth cutoffs $\psi_{k,j}: \mathbb{R}^n \to [0, 1]$ ($1 \le j \le J_k$) supported in caps $C_{k,j} \subset A_k$ of dimensions $\delta_k^2 \times \delta_k \times \cdots \times \delta_k$, with $\mathcal{L}^n(C_{k,j}) \asymp \delta_k^{n+1}$ and cap count \begin{align*} J_k \asymp \delta_k^{-(n-1)} = 2^{k(n-1)/2}. \end{align*} The associated cap-multiplier operators are \begin{align*} T_{k,j} : \mathcal{S}(\mathbb{R}^n) &\to \mathcal{S}'(\mathbb{R}^n), \\ f &\mapsto \mathcal{F}^{-1}(m_k\, \psi_{k,j}\, \widehat{f}), \end{align*} with $T_k = \sum_{j=1}^{J_k} T_{k,j}$. By the standard wave-packet description (Stein, *Harmonic Analysis*, Princeton University Press 1993, Chapter IX §5), each $T_{k,j} f$ is essentially concentrated on a dual *plate* $P_{k,j}$ of dimensions $\delta_k^{-2} \times \delta_k^{-1} \times \cdots \times \delta_k^{-1}$ oriented along the normal $\theta_{k,j} \in S^{n-1}$ to the sphere at the cap centre, with Schwartz-tail decay outside. [claim:Kakeya-endpoint per-shell bound (cited)] Granting the Kakeya maximal conjecture in $\mathbb{R}^n$, for every $\varepsilon > 0$ there exists $C_\varepsilon > 0$ such that for every $k \ge 0$, \begin{align*} \|T_k\|_{L^{p_{\mathrm{Kak}}}(\mathbb{R}^n) \to L^{p_{\mathrm{Kak}}}(\mathbb{R}^n)} \le C_\varepsilon\, 2^{-k\lambda}\, 2^{k\varepsilon}, \end{align*} where $p_{\mathrm{Kak}} = 2n/(n-1)$. (The Bochner--Riesz critical exponent at $p_{\mathrm{Kak}}$ is $\lambda(p_{\mathrm{Kak}}) = \max(0,\, n|1/p_{\mathrm{Kak}} - 1/2| - 1/2) = \max(0, 0) = 0$, so the gain factor reduces to $2^{-k\lambda}$.) [/claim] [proof] This is the content of Stein, *Harmonic Analysis*, Princeton University Press 1993, Chapter IX §5, Theorem 5.5 ("the maximal Kakeya operator on $L^n(S^{n-1})$ controls the Bochner--Riesz operator at the critical exponent $p_{\mathrm{Kak}}$"); see also Tao--Vargas, *Geom. Funct. Anal.* **10** (2000), §3 (the bilinear formulation), and Carbery, "The boundedness of the maximal Bochner--Riesz operator on $L^4(\mathbb{R}^2)$," *Duke Math. J.* **50** (1983), 409--416 (the original reduction in $\mathbb{R}^2$). The hypotheses verified are: (i) $T_k$ is the Fourier multiplier with symbol $m_k$ as above; (ii) $\|m_k\|_\infty \asymp 2^{-k\lambda}$; (iii) the Kakeya maximal conjecture is granted. The conclusion is the displayed inequality. The structure of the cited argument is: the wave-packet decomposition of $T_k f$ gives a sum over plate operators; the plate-localised pointwise bound has amplitude $\asymp 2^{-k\lambda}$ times a plate-mean of $f$; summing over the $J_k$ plates and using the bounded-overlap property reduces the sum to a Kakeya tube-sum estimate on the rescaled tubes (after parabolic rescaling at scale $\delta_k$); the Kakeya maximal hypothesis yields the tube-sum bound with a $\delta_k^{-\varepsilon} = 2^{k\varepsilon/2}$ loss; the parabolic-rescaling Jacobian factor on $L^{p_{\mathrm{Kak}}}$ combines with the volume gain from cap-to-plate to produce the stated bound. The full Schwartz-tail bookkeeping of the wave-packet decomposition is the technical content of Stein 1993 Ch IX §5. [/proof] [/step]

[step:Riesz--Thorin interpolation between $L^2$ and $L^{p_{\mathrm{Kak}}}$] **Compute $\lambda(p)$ on $[2, p_{\mathrm{Kak}}]$.** For $p \in [2, p_{\mathrm{Kak}}]$, we have $1/p \in [(n-1)/(2n), 1/2]$ and $|1/p - 1/2| = 1/2 - 1/p \in [0, 1/(2n)]$. Hence $n\,|1/p - 1/2| - 1/2 \in [-1/2, 0]$, all non-positive. Adopting the convention $\lambda(p) := \max(0,\, n|1/p - 1/2| - 1/2)$ (which is the standard Bochner--Riesz critical exponent — non-negative by construction, and the conjectured threshold), we have $\lambda(p) = 0$ for $p \in [2, p_{\mathrm{Kak}}]$. In particular $\lambda(p_{\mathrm{Kak}}) = 0$. So the Step 3 cited bound is $\|T_k\|_{L^{p_{\mathrm{Kak}}} \to L^{p_{\mathrm{Kak}}}} \le C_\varepsilon\, 2^{-k\lambda + k\varepsilon}$ (the $\lambda(p_{\mathrm{Kak}})$ term is zero), and Step 2 gives $\|T_k\|_{L^2 \to L^2} \le C\, 2^{-k\lambda}$. Apply the [Riesz--Thorin interpolation theorem](/theorems/3172) to the linear operator $T_k$ between the pairs $(L^2, L^2)$ and $(L^{p_{\mathrm{Kak}}}, L^{p_{\mathrm{Kak}}})$. Hypotheses verified: (i) $T_k$ is linear (as a Fourier multiplier); (ii) $T_k$ is bounded on $L^2$ and on $L^{p_{\mathrm{Kak}}}$ with the bounds above; (iii) for $p \in [2, p_{\mathrm{Kak}}]$ there exists $\theta \in [0, 1]$ with $1/p = (1-\theta)/2 + \theta/p_{\mathrm{Kak}}$. Solving: $\theta = (1/2 - 1/p)/(1/2 - 1/p_{\mathrm{Kak}}) = 2n\,(1/2 - 1/p) = n(1 - 2/p)$. Riesz--Thorin gives \begin{align*} \|T_k\|_{L^p \to L^p} \le \|T_k\|_{L^2 \to L^2}^{1-\theta}\, \|T_k\|_{L^{p_{\mathrm{Kak}}} \to L^{p_{\mathrm{Kak}}}}^\theta \le C_\varepsilon\, 2^{-k\lambda(1-\theta)}\, 2^{(-k\lambda + k\varepsilon)\theta} = C_\varepsilon\, 2^{-k\lambda + k\varepsilon\theta}. \end{align*} Absorbing $\theta \le 1$ into the constant, \begin{align*} \|T_k\|_{L^p \to L^p} \le C_\varepsilon\, 2^{-k\lambda + k\varepsilon}\, \|f\|_{L^p}. \end{align*} Since $\lambda(p) = 0$ on $[2, p_{\mathrm{Kak}}]$, this is equivalent to $\|T_k\|_{L^p \to L^p} \le C_\varepsilon\, 2^{-k(\lambda - \lambda(p)) + k\varepsilon}$, the form needed for Step 5. [/step]

[step:Sum the dyadic pieces using $\lambda > \lambda(p)$] Combine the bound on $\|T_k f\|_{L^p}$ with the triangle inequality: \begin{align*} \|T^\lambda f\|_{L^p(\mathbb{R}^n)} \le \sum_{k=0}^\infty \|T_k f\|_{L^p} \le C_\varepsilon\, \|f\|_{L^p} \sum_{k=0}^\infty 2^{-k(\lambda - \lambda(p) - \varepsilon)}. \end{align*} The geometric series converges if and only if $\lambda - \lambda(p) - \varepsilon > 0$. Since $\lambda > \lambda(p)$ by hypothesis, choose $\varepsilon \in (0, \lambda - \lambda(p))$ — concretely $\varepsilon := (\lambda - \lambda(p))/2$ — so the geometric series sums to $(1 - 2^{-(\lambda - \lambda(p))/2})^{-1}$. Hence \begin{align*} \|T^\lambda f\|_{L^p(\mathbb{R}^n)} \le \frac{C_\varepsilon}{1 - 2^{-(\lambda - \lambda(p))/2}}\, \|f\|_{L^p(\mathbb{R}^n)} =: C(\lambda, p, n)\, \|f\|_{L^p(\mathbb{R}^n)}. \end{align*} The constant $C(\lambda, p, n)$ depends on the Kakeya-maximal constant from the cited per-shell bound (Step 3), on the cutoff $\chi$ via the cap tiling, on $\lambda(p)$, and on $n$. This is the Bochner--Riesz bound for $\lambda > \lambda(p)$ in the range $p \in [2, p_{\mathrm{Kak}}]$. The dual range $p \in [p_{\mathrm{Kak}}', 2]$ follows by the self-duality $T^\lambda = (T^\lambda)^*$ (the multiplier $m_\lambda$ is real and even). [/step]