The Bochner–Riesz conjecture in dimensions $d\ge 3$: sharp $L^p$-boundedness of the Bochner–Riesz means and the equivalent uniform local decay estimate

Problem record for the open question left by Hytönen–Lappas (arXiv:2006.15858). The Bochner–Riesz conjecture predicts that for $0<\kappa<(d-1)/2$ the Bochner–Riesz means $B^\kappa$, the Fourier multiplier with symbol $(1-|\xi|^2)_+^\kappa$, are bounded on $L^p(\mathbb{R}^d)$ exactly for $p\in(2d/(d+1+2\kappa),\,2d/(d-1-2\kappa))$; equivalently, the multipliers $S_\tau$ localized to a $\tau$-neighborhood of the unit sphere satisfy the uniform decay estimate $\|S_\tau\|_{L^p\to L^p}\le C_\epsilon\tau^{-\epsilon}$ for all $0<\tau<1$ on the full range $2d/(d+1)<p<2d/(d-1)$. The necessity of the range and the case $d=2$ (Carleson–Sjölin) are classical; in every dimension $d\ge 3$ the conjecture remains open, the best partial ranges being due to Lee, Bourgain–Guth, and Guth–Hickman–Iliopoulou, and the weighted boundedness and compactness results built on the decay estimate (Lacey–Mena–Reguera; Hytönen–Lappas) remain conditional on it. A resolution would imply the Fourier restriction conjecture.

Background

The question studied here comes from the summability theory of Fourier integrals. For a function $f$ on $\mathbb{R}^d$, truncating its Fourier transform to the unit ball gives the ball multiplier, whose $L^p$ boundedness was famously disproved for every $p\ne2$ in dimensions $d\ge2$ by C. Fefferman (1971). The Bochner–Riesz means are the smoothed truncations that restore summability: for a summability index $\kappa>0$, $B^\kappa$ is the Fourier multiplier operator — an operator of the form $\widehat{Tf}(\xi)=m(\xi)\hat f(\xi)$ — with symbol $m(\xi)=(1-|\xi|^2)_+^\kappa$, where $t_+=\max(t,0)$. Uniform $L^p$ bounds for the dilated means are equivalent to convergence of the means to $f$ in $L^p$, so the sharp $p$-range of boundedness is the sharp $p$-range of Fourier summability; the threshold is governed by the Gaussian curvature of the unit sphere, across which the symbol decays at rate $\kappa$.

The necessity of the boundedness range is classical (from Fefferman's counterexample and its refinements): $B^\kappa$ can be bounded on $L^p$ only if $\kappa>d\,|1/p-1/2|-1/2$. The Bochner–Riesz conjecture asserts that this necessary condition is sharp: for $0<\kappa<(d-1)/2$,

$$B^\kappa:L^p(\mathbb{R}^d)\mapsto L^p(\mathbb{R}^d)\quad\text{if and only if}\quad p\in\left(\frac{2d}{d+1+2\kappa},\,\frac{2d}{d-1-2\kappa}\right),$$

equivalently $|1/p-1/2|<(1+2\kappa)/(2d)$. For $d=2$ this is a theorem, proved by Carleson–Sjölin (1972) with an alternative proof by Córdoba (1979). For $d\ge3$ only subranges are known. Writing $q=\max(p,p')$, where $p'$ denotes the Hölder conjugate of $p$, the classical $L^2$ restriction theorem for the sphere (Tomas, 1975) combined with summability arguments gives the conjectured bound for $q\ge2(d+1)/(d-1)$; Lee (2004) extended this to $q\ge2(d+2)/d$ using sharp bilinear restriction estimates; Bourgain–Guth (2011) reached $q>2(4d+3)/(4d-3)$, $q>(2d+1)/(d-1)$, and $q>4(d+1)/(2d-1)$ for $d\equiv0,1,2\pmod3$ respectively; Guth–Hickman–Iliopoulou (2019), via sharp $L^p$ estimates for Hörmander-type oscillatory integral operators (operators $T_\lambda f(x)=\int e^{i\lambda\varphi(x,y)}a(x,y)f(y)\,dy$ with non-degenerate phase, the operators introduced into this problem by Carleson and Sjölin), improved the bounds for $d\ge4$. All known ranges remain strictly inside the conjectured one: in $d=3$ boundedness is known only for $q>10/3$ where the conjecture predicts $q>3$.

The conjecture has an equivalent single-scale form. Let $\chi$ be a Schwartz cutoff function with $\mathbf 1_{[-1/4,1/4]}\le\chi\le\mathbf 1_{[-1/2,1/2]}$ and let $S_\tau$ be the Fourier multiplier with symbol $\chi((|\xi|-1)/\tau)$, that is, a smooth restriction of $f$ to frequencies in a thin annulus of width $\tau$ around the unit sphere. Lacey–Mena–Reguera (2019) record the equivalent conjecture that for every $p$ with $2d/(d+1)<p<2d/(d-1)$ and every $\epsilon>0$,

$$\|S_\tau\|_{L^p(\mathbb{R}^d)\to L^p(\mathbb{R}^d)}\le C_\epsilon\,\tau^{-\epsilon},\qquad 0<\tau<1.$$

The equivalence goes through the classical decomposition $B^\kappa=T^0+\sum_{i\ge1}2^{-i\kappa}\operatorname{Dil}_{1-2^{-i}}S_{2^{-i}}$, where $T^0$ is a Fourier multiplier with Schwartz symbol supported near the origin and $\operatorname{Dil}_s$ is the dilation $f(x)\mapsto f(x/s)$: decay of $S_\tau$ with loss $\epsilon$ at exponent $p$ sums to boundedness of $B^\kappa$ at $p$ for every $\kappa>\epsilon$. The decay formulation makes visible why the problem is hard: it is a Fourier restriction statement for the sphere. Indeed Tao (1999) proved that the Bochner–Riesz conjecture implies the restriction conjecture for the sphere — the conjectured sharp estimates for the extension operator that maps a function on the sphere to the inverse Fourier transform of its product with surface measure — which in turn drives the Kakeya conjecture on the overlap of thin tubes pointing in different directions. The strongest available methods — multilinear restriction (Bennett–Carbery–Tao, 2006), the broad–narrow decomposition, polynomial partitioning (partitioning physical space by zero sets of polynomials, as in Guth, 2018), and the decoupling method (Bourgain–Demeter, 2015) — improve the known ranges in restricted dimension or exponent regimes but stop strictly short of the conjectured boundary in every $d\ge3$.

The present record arises from the weighted endpoint of this theory. A Muckenhoupt $A_p$ weight is a nonnegative locally integrable function $w$ on $\mathbb{R}^d$ with $(\fint_Q w)(\fint_Q w^{1-p'})^{p-1}\le C$ uniformly over cubes $Q$ ($\fint_Q$ denoting the average over $Q$), and $RH_r$ denotes the reverse Hölder class, $\left(\fint_Q w^r\right)^{1/r}\le C\fint_Q w$. For $b$ in $\mathrm{BMO}$ (bounded mean oscillation), the commutator of $b$ with an operator $T$ is $[b,T]f=b\,Tf-T(bf)$. Lacey–Mena–Reguera (2019) proved sparse (i.e. domination by sums over sparse families of cubes, from which weighted bounds follow), and consequently weighted, bounds for Bochner–Riesz multipliers: in $d=2$ unconditionally, but in $d\ge3$ only under the hypothesis that the decay estimate above holds at some exponent $1<p_0<2$, the resulting range of $p$ and the weight classes $A_s\cap RH_r$ depending on this $p_0$. Hytönen–Lappas (2020), extending the classical weighted extrapolation machinery to compactness of operators (an operator is compact if it maps bounded sets to sets with compact closure), used exactly this input to deduce weighted compactness of commutators of Bochner–Riesz multipliers with $\mathrm{CMO}$ symbols (the closure of compactly supported smooth functions in $\mathrm{BMO}$) in $d\ge3$ — again conditional on the same decay hypothesis. The unresolved analytic core beneath the entire weighted chain is the decay conjecture, hence the Bochner–Riesz conjecture, in dimensions $d\ge3$.

Problem Statement

Determine, for each dimension $d\ge3$, whether the Bochner–Riesz conjecture holds: is it true that for every $0<\kappa<(d-1)/2$ the multiplier operator $B^\kappa$ with symbol $(1-|\xi|^2)_+^\kappa$ is bounded on $L^p(\mathbb{R}^d)$ for all

$$p\in\left(\frac{2d}{d+1+2\kappa},\,\frac{2d}{d-1-2\kappa}\right),$$

equivalently for all $p$ with $|1/p-1/2|<(1+2\kappa)/(2d)$? The necessity of this range is classical and the case $d=2$ is settled (Carleson–Sjölin), so the open content is sufficiency in $d\ge3$. An equivalent target is the uniform local decay estimate for the thin-annulus multipliers $S_\tau$ with symbol $\chi((|\xi|-1)/\tau)$, where $\chi$ is any Schwartz cutoff with $\mathbf 1_{[-1/4,1/4]}\le\chi\le\mathbf 1_{[-1/2,1/2]}$:

$$\|S_\tau\|_{L^p(\mathbb{R}^d)\to L^p(\mathbb{R}^d)}\le C_\epsilon\,\tau^{-\epsilon},\qquad 0<\tau<1,$$

required for every $\epsilon>0$ and every $p$ in the full range $2d/(d+1)<p<2d/(d-1)$. Resolving either formulation in $d\ge3$ settles the other; in particular, it would remove the dependence on an admissible $p_0$ — the hypothesis that the decay estimate holds for some $1<p_0<2$ — from the weighted boundedness and weighted compactness results for Bochner–Riesz multipliers and their commutators (Lacey–Mena–Reguera, Theorem 6.1; Hytönen–Lappas).

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