Tao's maximal Bochner–Riesz conjecture for $1\le p<2$: almost everywhere convergence of Bochner–Riesz means at the critical order

For $f\in L^p(\mathbb{R}^n)$, the Bochner–Riesz means $T^\lambda_t f$ (the Fourier multiplier with symbol $(1-|\xi|^2/t^2)_+^\lambda$) recover $f$ pointwise as $t\to\infty$ only above a critical summability order $\lambda$. For $p\ge 2$ the sharp range is known (Carbery–Rubio de Francia–Vega, 1988). For $1\le p<2$, Tao conjectured (1998) that the maximal operator $T^\lambda_* f=\sup_{t>0}|T^\lambda_t f|$ is bounded on $L^p(\mathbb{R}^n)$ exactly when $\lambda>\max\{0,(2n-1)/(2p)-n/2\}$, the threshold being known to be best possible from below. Despite a chain of partial results culminating in Li and Wu's 2024 proof of the planar case for $5/3\le p\le 2$ (the first sharp result with $\lambda$ near 0), the conjecture remains open for $1\le p<5/3$ in the plane and in all dimensions $n\ge 3$; for $n\ge 4$ no non-trivial positive result is known at all. The problem is to prove or refute the conjecture on this remaining range.

Background

The Bochner–Riesz means are the classical radial summation method for Fourier integrals on $\mathbb{R}^n$: for a summability order $\lambda>0$ and scale $t>0$,

$$T^{\lambda}_{t}f(x):=(2\pi)^{-n}\int_{|\xi|\le t}\Big(1-\frac{|\xi|^{2}}{t^{2}}\Big)^{\lambda}_{+}\widehat{f}(\xi)\,e^{ix\cdot\xi}\,d\xi,$$

i.e. $T^\lambda_t$ is the Fourier multiplier (an operator acting by multiplying the Fourier transform $\widehat{f}$ by a fixed function) with symbol $(1-|\xi|^2/t^2)_+^\lambda$, where $(\cdot)_+$ denotes the positive part. These means were introduced by Bochner as a higher-dimensional analogue of Abel/Cesàro summation of one-dimensional Fourier series: they average the spectrum in the ball of radius $t$ with a smooth radial profile, and the question is how much smoothing $\lambda$ is needed for the means to converge to $f$ as $t\to\infty$. At $\lambda=0$ the operator truncates to the disk multiplier, and Fefferman's ball-multiplier theorem shows this already fails on $L^p$ for every $p\ne 2$ in $n\ge 2$, so the question is meaningful only for $\lambda>0$.

For pointwise convergence, Stein's maximal principle (Stein, 1961) reduces the question to boundedness of the maximal Bochner–Riesz operator $T^{\lambda}_{*}f:=\sup_{t>0}|T^{\lambda}_{t}f|$: convergence $T^\lambda_t f\to f$ almost everywhere for every $f\in L^p(\mathbb{R}^n)$ is equivalent to an $L^p$ bound for $T^\lambda_*$ (in the range $p<2$; see the formulation in Li–Wu, 2024, after Tao). For $p\ge 2$ the problem is completely solved: Carbery, Rubio de Francia and Vega (1988) proved almost everywhere convergence in the sharp range of $\lambda$.

The remaining case $1\le p<2$ — functions rougher than $L^2$, for which the radial Fourier summability threshold is genuinely different — was formulated as a conjecture by Tao (1998): $T^\lambda_*$ extends to a bounded operator on $L^p(\mathbb{R}^n)$ whenever

$$\lambda>\max\Big\{0,\ \frac{2n-1}{2p}-\frac{n}{2}\Big\},$$

and this threshold is sharp, since Tao's rescaled bump-packet example disproves the maximal bound below it (the critical order $\lambda_c=\max\{0,(2n-1)/(2p)-n/2\}$ vanishes at $p=2-1/n$ and grows to $(n-1)/2$ as $p\downarrow 1$; the hardest regime is $\lambda$ close to $0$ near $p=2-1/n$). This conjecture is the last open piece of Stein's program on almost everywhere summability of Fourier integrals, and it has been a driving force behind modern restriction and decoupling theory — in particular the Bourgain–Demeter $\ell^2$-decoupling theorem (2015), which controls $L^p$ norms of functions with Fourier support near a curved hypersurface by decomposing the support into small caps, is the engine of the strongest partial results.

The known partial results in the plane, prior to 2024, established the maximal bound only on a quadrilateral region strictly inside the conjectured triangle in the $(1/p,\lambda)$ plane: Tao (2002) via an $L^{10/7}$ estimate, Li–Wu (2020) via an $L^{18/13}$ estimate using decoupling, and Gan–Wu (2024) and Kim (2025) via weighted $\ell^p$-decoupling inequalities, which also gave the first beyond-interpolation result in $\mathbb{R}^3$ and improved sufficient conditions for $1<p<2$ in dimensions 2 and 3 (still far from sharp). Li and Wu (2024, revised 2026) then proved the conjecture for $n=2$ in the range $5/3\le p\le 2$ — the first sharp non-trivial result for $p<2$ with $\lambda$ near $0$, obtained through quantitative weighted $L^2$ estimates over unions of unit balls — so that in the plane the means converge almost everywhere for every $f\in L^p(\mathbb{R}^2)$ and every $\lambda>0$ when $5/3\le p\le 2$. They explicitly state that the remaining planar region (their Region B, $1\le p<5/3$) is open, and that in dimensions $n\ge 4$ no non-trivial positive result is known even for $\lambda$ away from zero.

Problem Statement

Prove or disprove Tao's maximal Bochner–Riesz conjecture for $p<2$: for every dimension $n\ge 2$ and every $1\le p<2$, the maximal Bochner–Riesz operator

$$T^{\lambda}_{*}f(x)=\sup_{t>0}\Big|(2\pi)^{-n}\int_{|\xi|\le t}\Big(1-\frac{|\xi|^{2}}{t^{2}}\Big)^{\lambda}_{+}\widehat{f}(\xi)\,e^{ix\cdot\xi}\,d\xi\Big|$$

extends to a bounded operator on $L^p(\mathbb{R}^n)$ whenever $\lambda>\max\{0,(2n-1)/(2p)-n/2\}$; equivalently, by Stein's maximal principle, $T^{\lambda}_{t}f\to f$ almost everywhere for every $f\in L^p(\mathbb{R}^n)$ exactly in this range of $\lambda$ (the threshold being already known to be sharp from below, by Tao's example). The segment $n=2$, $5/3\le p\le 2$ is already proved by Li and Wu and need not be reproved; the surviving open core is the planar region $1\le p<5/3$ together with the full conjecture in all dimensions $n\ge 3$, where for $n\ge 4$ not even a non-sharp positive result is currently known. A valid refutation must exhibit, for some $n\ge 2$, $1\le p<2$ and $\lambda$ strictly above the conjectured threshold, an $L^p$ function for which the maximal bound (equivalently, almost everywhere convergence) fails.

References

1. Xiaochun Li and Shukun Wu, "On almost everywhere convergence of planar Bochner-Riesz means", arXiv:2407.20887 (2024; v3 revised April 2026), https://arxiv.org/abs/2407.20887
2. Jongchon Kim, "Weighted decoupling estimates and the Bochner-Riesz means", Forum of Mathematics, Sigma 13 (2025) e167, DOI 10.1017/fms.2024.96, https://doi.org/10.1017/fms.2024.96 (arXiv:2406.03741, https://arxiv.org/abs/2406.03741)
3. Adam D. Horwich and Alessio Martini, "Almost everywhere convergence of Bochner–Riesz means on Heisenberg-type groups", Journal of the London Mathematical Society 103 (2021), no. 3, 1066–1119, DOI 10.1112/jlms.12401, https://doi.org/10.1112/jlms.12401 (arXiv:1908.04049, https://arxiv.org/abs/1908.04049)
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7. Shengwen Gan and Shukun Wu, "A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem", arXiv:2403.05017 (2024), https://arxiv.org/abs/2403.05017
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