Math Conjectures

Bochner–Riesz conjecture

OPENLandmarkConjectureProposed c. 1971 · Standard version

Canonical statement

Let n≥3n\ge3, 1<p<∞1<p<\infty, and
δ>max⁡ ⁣{n∣1p−12∣−12, 0}. \delta>\max\!\left\{n\left|\frac1p-\frac12\right|-\frac12,\,0\right\}.
For every Schwartz function ff and R>0R>0, define
SRδf^(ξ)=(1−∣ξ∣2R2)+δf^(ξ),a+=max⁡{a,0}. \widehat{S_R^\delta f}(\xi) =\left(1-\frac{|\xi|^2}{R^2}\right)_+^\delta\widehat f(\xi), \qquad a_+=\max\{a,0\}.
Then sup⁡R>0∥SRδf∥Lp(Rn)≤Cn,p,δ∥f∥Lp(Rn)\sup_{R>0}\|S_R^\delta f\|_{L^p(\mathbb R^n)} \le C_{n,p,\delta}\|f\|_{L^p(\mathbb R^n)}.
View source LaTeX
Let \(n\ge3\), \(1<p<\infty\), and \[ \delta>\max\!\left\{n\left|\frac1p-\frac12\right|-\frac12,\,0\right\}. \] For every Schwartz function \(f\) and \(R>0\), define \[ \widehat{S_R^\delta f}(\xi) =\left(1-\frac{|\xi|^2}{R^2}\right)_+^\delta\widehat f(\xi), \qquad a_+=\max\{a,0\}. \] Then \(\sup_{R>0}\|S_R^\delta f\|_{L^p(\mathbb R^n)} \le C_{n,p,\delta}\|f\|_{L^p(\mathbb R^n)}\).
ANAL-FA-008:[Stein1993HarmonicAnalysis][GuthHickmanIliopoulou2019Oscillatory][LiWu2021BochnerRieszRevisited][GanWu2025WeightedDecoupling]
Analysis and functional analysis

Notes

The Bochner–Riesz means SRδS_R^\delta smooth out the sharp frequency cutoff of the ball by the multiplier (1−∣ξ∣2/R2)+δ(1-|\xi|^2/R^2)_+^\delta. The conjecture, whose standard form emerged in early-1970s harmonic analysis, asserts that these means are bounded on Lp(Rn)L^p(\mathbb R^n), uniformly in RR, whenever δ>max⁡{n∣1/p−1/2∣−1/2, 0}\delta>\max\{n|1/p-1/2|-1/2,\,0\}, a threshold dictated by kernel decay and standard counterexamples. It quantifies how much smoothing is needed for spherical summation of Fourier integrals to behave well in LpL^p [Stein1993HarmonicAnalysis].

In the plane the conjectured range is a theorem. For n≥3n\ge3, substantial subranges are known, largely as consequences of restriction and oscillatory-integral estimates; sharp bounds obtained via polynomial partitioning represent the state of the art in that direction [GuthHickmanIliopoulou2019Oscillatory]. An older approach to the problem has also been revisited [LiWu2021BochnerRieszRevisited], and weighted decoupling estimates have recently been brought to bear on the means [GanWu2025WeightedDecoupling].

The problem is closely tied to the restriction and Kakeya conjectures. In every dimension n≥3n\ge3 part of the conjectured range remains unproved, and the conjecture remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.