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on 3D critical Bochner–Riesz exponent ($C_{77}$)

The statement

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]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.