Bochner–Riesz conjecture

A function can be reconstructed from waves of different frequencies. When waves above a spherical frequency boundary are discarded, can a taper near the boundary keep the reconstructed function’s overall size under control, uniformly as the cutoff radius changes? For taper exponent one and cutoff radius two, a wave of frequency length one gets weight $3/4$, while lengths two and three get zero. These are Bochner–Riesz means. Size is measured by an Lp norm, which averages powers of the function’s magnitude. The page asks for a uniform Lp bound in the specified conjectured exponent range in dimensions three and higher, not merely convergence at individual points.[1] [2]

Can a tapered spherical selection of frequencies avoid an unbounded increase in the Lp size of a function as its radius changes? Write $S_R^\delta$ for the Fourier multiplier with symbol $m_R^\delta(\xi)=(1-|\xi|^2/R^2)_+^\delta$. The subscript plus means zero below zero. For $R=2$, $\delta=1$, frequency lengths one and two receive weights $3/4$ and zero. The target is $\sup_{R>0}\|S_R^\delta\|_{L^p\to L^p}<\infty$ on $\mathbb R^d$, with $d\ge3$, $2d/(d-1)\le p<\infty$ and $\delta>d(1/2-1/p)-1/2$. For instance, $d=3$, $p=4$ gives $\delta>1/4$; verifying that arithmetic is not a proof of the operator bound.[1] [2]

The classical boundedness result is known in two dimensions, and higher-dimensional work covers restricted exponent ranges. The question here is the full stated range in dimensions at least three. Almost-everywhere convergence, even in that range, only concerns values at almost every point and does not give the requested uniform Lp operator bound. The September preprint changes to weighted estimates and other convex cutoff shapes; its reported counterexamples do not refute the spherical unweighted target.[1] [2] [3]

The September preprint reports counterexamples for C^k-convex weighted-L² variants, changing the geometry and weight. These claims do not refute this unweighted spherical conjecture.[3]

Waves that are small individually can reinforce one another and create large peaks. A curved frequency boundary influences that interference, so the question links the geometry of selected frequencies with the total size of the reconstructed function. Lp control measures more than success at individual points: it controls the aggregate magnitude of those peaks as the cutoff changes. The same oscillatory estimates connect to Fourier restriction, which studies a function’s frequency data on curved surfaces.[1] [2]

A multiplier’s support has geometric structure, but its boundedness depends on cancellation in the resulting oscillatory integrals. Bochner–Riesz is a sharp test of how curved frequency geometry affects that cancellation and the size of reconstructed functions. The cited restriction approach connects these operator bounds with estimates for Fourier data on curved surfaces. Uniform Lp control also differs materially from almost-everywhere convergence: the latter need not supply an aggregate norm estimate across cutoff scales. The weighted and nonspherical variants illustrate why the exact measure, geometry and exponent assumptions matter.[1] [2]