The Bochner–Riesz multiplier conjecture in $\mathbb{R}^n$ for $n\ge3$

The Bochner–Riesz conjecture predicts the sharp $L^p$ summability range of the Bochner–Riesz means, the Fourier multipliers $(1-|\xi|^2)_+^\alpha$ on $\mathbb{R}^n$: the operator $S^\alpha$ should be bounded on $L^p$ exactly when $\alpha$ exceeds the critical index $\alpha(p)=\max(n|1/2-1/p|-1/2,\,0)$. The necessity direction is classical and dimension $n=2$ is fully resolved (Carleson–Sjölin), but in every dimension $n\ge3$ the conjecture remains open: it is currently proved only for $\max\{p,p/(p-1)\}\ge3.25$ in $n=3$, beyond $2+200/251\approx2.7968$ in $n=4$, and beyond the best verified Fourier restriction ranges in $n\ge5$, leaving open the entire band $2n/(n+1)\le p\le 2n/(n-1)$, $p\ne2$, where the critical index vanishes and boundedness must hold for every $\alpha>0$.

Background

Fix $n\ge2$ and let $\hat f(\xi)=\int_{\mathbb{R}^n}f(x)e^{-2\pi ix\cdot\xi}\,dx$ be the Fourier transform. A Fourier multiplier is the operator $m(D)$ defined by damping the frequencies of $f$ by a bounded function $m$ and transforming back: $m(D)f(x)=\int e^{2\pi ix\cdot\xi}m(\xi)\hat f(\xi)\,d\xi$. Plain spherical cut-off of the Fourier inversion integral fails on $L^p$ for $p\ne2$ — most dramatically, Fefferman's 1971 solution of the ball multiplier problem shows the ball indicator multiplier ($m=\mathbf{1}_{|\xi|\le1}$) is unbounded on $L^p(\mathbb{R}^n)$ for every $p\ne2$, $n\ge2$. The Bochner–Riesz means, introduced by Bochner in the 1930s, restore summability by damping smoothly: their multiplier of order $\alpha\ge0$ is $(1-|\xi|^2)_+^\alpha$, where $t_+=\max(t,0)$; larger $\alpha$ gives a smoother vanishing at the unit sphere.

The question is: on which $L^p$ is the damped spherical summation bounded? Stationary-phase asymptotics (the standard approximation scheme for oscillatory integrals near critical points of the phase) of the convolution kernel (an oscillatory Bessel-type kernel decaying like $|x|^{-(n+1)/2-\alpha}$) classically imply that the means are necessarily unbounded whenever $\alpha\le\alpha(p):=\max(n|1/2-1/p|-1/2,\,0)$ and $p\ne2$; this threshold $\alpha(p)$ is the critical index. The Bochner–Riesz conjecture asserts the converse: $S^\alpha$ is bounded on $L^p$ for every $\alpha>\alpha(p)$. Equivalently, at fixed order $0<\alpha\le(n-1)/2$, boundedness should hold exactly for $\frac{2n}{n+1+2\alpha}<p<\frac{2n}{n-1-2\alpha}$ (for $\alpha>(n-1)/2$ the kernel is integrable and all $p$ are fine; for $p=2$ all $\alpha\ge0$ work). Note that $\alpha(p)=0$ on the whole band $\frac{2n}{n+1}\le p\le\frac{2n}{n-1}$, so there the conjecture demands boundedness for every $\alpha>0$ — arbitrarily close to the undamped, unbounded ball multiplier.

In $n=2$ the conjecture is a theorem of Carleson and Sjölin (1972), with alternative proofs by Hörmander (1973) and Fefferman (1973). Their key step — now called the Carleson–Sjölin reduction — rewrites the dyadic pieces of $S^\alpha$ as oscillatory integral operators $T^\lambda f(x)=\int e^{i\lambda\phi(x,y)}a(x,y)f(y)\,dy$ with curved phase $\phi$, so that $L^p$ bounds for such operators (obtained from cancellation between wave packets) yield the conjectured multiplier bounds. In $n\ge3$ only partial ranges are known. Restriction-based arguments came first: Fefferman (1973) observed that Fourier restriction estimates on the sphere (bounds for $\widehat{f\,d\sigma}$, the Fourier transform of a density $f$ with respect to surface measure $d\sigma$ on the unit sphere $S^{n-1}\subset\mathbb{R}^n$) imply Bochner–Riesz bounds, Tomas (1975) supplied such estimates by the $TT^*$ method (bounding an operator $T$ through the composition $TT^*$ of $T$ with its adjoint, whose kernel is positive and therefore easier to estimate), and Bourgain (1991) improved the range using Besicovitch-type maximal function estimates from Kakeya-type geometry — the incidence geometry of unions of thin tubes pointing in different directions. Lee (2004) and Bourgain–Guth (2011) advanced further via bilinear/multilinear restriction and the broad–narrow analysis of where $|T^\lambda f|$ can concentrate (mass either spreads broadly over directions or concentrates narrowly on algebraic varieties). Guth–Hickman–Iliopoulou (2019) reached the sharp $L^p$ range for positive-definite-phase oscillatory integral operators using polynomial partitioning (partitioning physical space by the zero set of a polynomial to locate and exhaust concentration), which by the same reduction gives the conjecture for $p\ge2\cdot\frac{3n+1}{3n-3}$ ($n$ odd) and $p\ge2\cdot\frac{3n+2}{3n-2}$ ($n$ even).

The problem is inseparable from the Fourier restriction conjecture for the sphere — the assertion that the extension operator $f\mapsto\widehat{f\,d\sigma}$ maps $L^\infty(S^{n-1})$ to $L^p(\mathbb{R}^n)$ for all $p>\frac{2n}{n-1}$: Tao (1999) proved that the Bochner–Riesz conjecture implies the restriction conjecture, Carbery (1992) proved the reverse implication for the paraboloid, and Guo–Oh–Wang–Wu–Zhang (2025) showed that the whole restriction toolbox transfers to Bochner–Riesz via a pseudo-conformal transformation, so the conjectured Bochner–Riesz range and the verified restriction ranges now move together. Currently the conjecture is established only for $\max\{p,p/(p-1)\}\ge3.25$ in $n=3$ (Wu; Guo–Oh–Wang–Wu–Zhang), $\max\{p,p/(p-1)\}>2+\frac{200}{251}\approx2.7968$ in $n=4$ (Borges–Chan–Chen–Liu–Xi–Zhan), and for $p$ beyond the best verified restriction range in $n\ge5$. No method reaches the endpoint $p=\frac{2n}{n-1}$, and for $2<p<\frac{2n}{n-1}$ the conjectured boundedness for all small $\alpha>0$ is open.

Problem Statement

Let $n\ge3$ and let $S^\alpha$ be the Bochner–Riesz operator on $\mathbb{R}^n$ of order $\alpha\ge0$,

$S^\alpha f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\,(1-|\xi|^2)_+^\alpha\,\hat f(\xi)\,d\xi ,$

where $\hat f(\xi)=\int_{\mathbb{R}^n}f(x)\,e^{-2\pi ix\cdot\xi}\,dx$ is the Fourier transform of $f$ and $t_+=\max(t,0)$ for real $t$. Define the critical index $\alpha(p)=\max\bigl(n\,\bigl|\tfrac12-\tfrac1p\bigr|-\tfrac12,\,0\bigr)$. The classical necessity argument gives unboundedness on $L^p$ for every $\alpha\le\alpha(p)$, $p\ne2$. The Bochner–Riesz conjecture asserts the converse:

$\text{for all } 1\le p\le\infty,\; p\ne2, \quad \|S^\alpha f\|_{L^p(\mathbb{R}^n)}\le C_{n,p,\alpha}\,\|f\|_{L^p(\mathbb{R}^n)} \quad\text{if and only if}\quad \alpha>\alpha(p).$

Equivalently, at fixed order $0<\alpha\le(n-1)/2$, $S^\alpha$ should be $L^p$-bounded exactly on the interval $\frac{2n}{n+1+2\alpha}<p<\frac{2n}{n-1-2\alpha}$ (for $\alpha>(n-1)/2$, boundedness should hold for all $p$; for $p=2$ it holds for all $\alpha\ge0$). The problem is to settle this conjecture in every dimension $n\ge3$ (it is a theorem for $n=2$). The open core is the full parameter region with $2<\max\{p,p/(p-1)\}$ below the currently verified thresholds — $3.25$ in $n=3$, $2+\frac{200}{251}$ in $n=4$, and the verified restriction range $p_{\mathrm{restr}}(n)$ in $n\ge5$ — in particular the entire band $\frac{2n}{n+1}\le p\le\frac{2n}{n-1}$, $p\ne2$, where $\alpha(p)=0$ and boundedness must be established for every $\alpha>0$, including arbitrarily close to the endpoint $p=\frac{2n}{n-1}$ that no current method reaches.

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