3D critical Bochner–Riesz exponent

Description of constant

In harmonic analysis, for $\lambda > 0$ let $T^\lambda$ denote the Bochner–Riesz operator on $\mathbb{R}^3$, initially defined for Schwartz functions $f \in \mathcal{S}(\mathbb{R}^3)$ by \(T^\lambda f(x) := \int_{\mathbb{R}^3} (1-\lvert \xi \rvert^2)_+^\lambda \widehat{f}(\xi)e^{ix\cdot \xi}\,d\xi,\) where $\widehat{f}$ denotes the Fourier transform of $f$ and $(t)_+ := \max{t,0}$. For $1 \le p \le \infty$, let $p’$ denote the Hölder conjugate exponent, defined by $1/p + 1/p’ = 1$ with the usual conventions $1’=\infty$ and $\infty’=1$. The Bochner–Riesz conjecture predicts that \(\lVert T^\lambda f\rVert_{L^p(\mathbb{R}^3)} \le C_{p,\lambda}\lVert f\rVert_{L^p(\mathbb{R}^3)}\) whenever \(\lambda > \lambda_{3,p} := \max\Bigl\{0, 3\Bigl\lvert \frac1p - \frac12\Bigr\rvert - \frac12\Bigr\}.\) [Wu2023-def-operator] [Wu2023-conj-formula]

We define \(C_{77} := \inf\Bigl\{\Gamma \ge 3 : \text{for every }1 \le p \le \infty\text{ with }\max\{p,p'\}\ge \Gamma\text{ and every }\lambda > \lambda_{3,p},\ T^\lambda \text{ extends boundedly on }L^p(\mathbb{R}^3)\Bigr\}.\)

Thus $C_{77}$ is the least symmetric exponent $\Gamma$ for which the conjectural $L^p$ range is established throughout the region $\max{p,p’}\ge \Gamma$. The conjectural value is $C_{77}=3$, and the best established range is \(3 \le C_{77} \le \frac{13}{4}.\) [Wu2023-conj-formula] [Wu2023-thm-3.25]

Known upper bounds

Known lower bounds

Additional comments and links

• Relation to Fourier restriction. Tao proved that the Bochner–Riesz conjecture implies the restriction conjecture, and recent work shows that after a pseudo-conformal transformation the modern restriction machinery applies directly to the Bochner–Riesz problem. [GOWWZ2025-restriction-link]

• A second proof of the current upper bound. The 2025 paper of Guo–Oh–Wang–Wu–Zhang recovers the three-dimensional $\frac{13}{4}$ range by a different and slightly simpler approach. [GOWWZ2025-r3-recover]

• Foundational references for earlier milestones. The Tomas $TT^*$ bound, Lee’s three-dimensional improvements, and Tao’s implication “Bochner–Riesz $\Rightarrow$ restriction” are standard landmarks repeatedly cited in modern surveys. [Wu2023-historical-ranges] [GOWWZ2025-restriction-link]

References

• [GOWWZ2025] Guo, Shaoming; Oh, Changkeun; Wang, Hong; Wu, Shukun; Zhang, Ruixiang. The Bochner–Riesz Problem: An Old Approach Revisited. Peking Mathematical Journal 8 (2025), 201–270. DOI: 10.1007/s42543-023-00082-4. arXiv PDF: arXiv:2104.11188. Google Scholar
  • [GOWWZ2025-restriction-link] loc: arXiv v1 PDF p.2, Introduction, paragraph beginning “Tao [Tao99] proved…” quote: “Tao [Tao99] proved that the Bochner-Riesz conjecture implies the restriction conjecture. Moreover, he mentioned in his paper that these two conjectures ‘are widely believed to be at least heuristically equivalent’. The information we would like to convey in the current paper is that, after applying the pseudo-conformal transformation (see (2.12) below), the recently developed techniques in the Fourier restriction literature apply equally well to the Bochner-Riesz problem.”
  • [GOWWZ2025-r3-recover] loc: arXiv v1 PDF pp.3–4, Introduction, Remark 1.4 quote: “Regarding the Bochner-Riesz problem in $n = 3$, recently Wu [Wu20] proved that the Bochner-Riesz conjecture holds for $p \ge p_{\mathrm{restr}} = 3.25$ when $n = 3$. His proof partially relies on some ideas from Wang [Wan18]. Our Theorem 1.3 recovers the result in [Wu20] via a quite different and a slightly simpler approach.”
• [Wu2023] Wu, Shukun. On the Bochner–Riesz operator in $\mathbb{R}^3$. Journal d’Analyse Mathématique 149 (2023), no. 2, 677–718. DOI: 10.1007/s11854-022-0263-y. arXiv PDF: arXiv:2008.13043. Google Scholar
  • [Wu2023-def-operator] loc: arXiv v2 PDF p.1, Introduction, equation (1.1) quote: “Recall that for $\lambda > 0$, the Bochner-Riesz multiplier of order $\lambda$ in $\mathbb{R}^n$ is defined by $T^\lambda f(x) = \int_{\mathbb{R}^n} (1-\lvert \xi \rvert^2)_+^\lambda \widehat{f}(\xi)e^{ix\cdot \xi}\,d\xi$.”
  • [Wu2023-conj-formula] loc: arXiv v2 PDF p.1, Introduction, Conjecture 1.1 and equation (1.3) quote: “Conjecture 1.1. (Bochner-Riesz) Assume $f \in L^p(\mathbb{R}^n)$ and $1 \le p \le \infty$. Then $\lVert T^\lambda f\rVert_p \le C_{p,\lambda}\lVert f\rVert_p$ for $\lambda > \lambda_{n,p}$, where the factor $\lambda_{n,p}$ is defined to be $\lambda_{n,p} = \max\Bigl{0, n\Bigl\lvert \frac1p - \frac12\Bigr\rvert - \frac12\Bigr}$.”
  • [Wu2023-historical-ranges] loc: arXiv v2 PDF p.1, Introduction, paragraph beginning “In higher dimensions, Tomas [17] proved…” quote: “In higher dimensions, Tomas [17] proved that the Bochner-Riesz conjecture is true when $\max{p, p/(p - 1)} \ge 2(n + 2)/(n - 1)$, via a $TT^*$ method. This result was improved by Bourgain [1] later in 1991, using new estimates for the Nikodym maximal function. After that, improvements have been made by several authors. See for instance, [20], [12], [2]. To the author’s knowledge, in $\mathbb{R}^3$, the best result so far is due to Lee [12] and [13], who proved that the Bochner-Riesz conjecture is true when $\max{p, p/(p - 1)} \ge 10/3$; In $\mathbb{R}^n$, $n \ge 4$, the best results are given by Guth, Hickman and Iliopoulou [9].”
  • [Wu2023-thm-3.25] loc: arXiv v2 PDF p.2, Introduction, Theorem 1.2 quote: “Theorem 1.2. Let $T^\lambda$ be the Bochner-Riesz operator defined in (1.1). Then the Bochner-Riesz conjecture (1.2) holds when $n = 3$, $\max{p, p/(p - 1)} \ge 3.25$.”

• [Tomas1975] Tomas, Peter A. A restriction theorem for the Fourier transform. Bulletin of the American Mathematical Society 81 (1975), no. 4, 477–478. Publisher page: Project Euclid. Google Scholar

• [Lee2004] Lee, Sanghyuk. Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators. Duke Mathematical Journal 122 (2004), no. 1, 205–232. DOI: 10.1215/S0012-7094-04-12217-1. Google Scholar

• [Lee2006] Lee, Sanghyuk. Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces. Journal of Functional Analysis 241 (2006), no. 1, 56–98. DOI: 10.1016/j.jfa.2006.05.011. Google Scholar

• [Tao1999] Tao, Terence. The Bochner-Riesz conjecture implies the restriction conjecture. Duke Mathematical Journal 96 (1999), no. 2, 363–375. DOI: 10.1215/S0012-7094-99-09610-2. Google Scholar

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.