Sharp Bochner–Riesz summability thresholds beyond single-singularity models: polygonal domains and multi-cone surfaces, and scaling-critical magnetic Schrödinger operators in dimensions $n\ge3$

The Bochner–Riesz means $S^\delta_\lambda=(1-L/\lambda^2)_+^\delta$ of a Laplace-type operator $L$ are smoothed spectral projections whose uniform $L^p$ boundedness is governed, in the settled models, by a sharp summability threshold $\delta^*(p)$. The threshold is known for the flat Laplacian on $\mathbb R^2$, for a single two-dimensional flat cone and infinite sectors (Jia–Zhang–Zheng 2025), and for the planar scaling-critical Aharonov–Bohm magnetic Schrödinger operator (Miao–Yan–Zhang 2024). The surviving open problem has two linked components sharing one objective — a sharp if-and-only-if threshold criterion for operators whose spectral measures contain genuinely global diffractive or magnetic structure: (A) the sharp Bochner–Riesz threshold on bounded planar polygonal domains (Dirichlet and Neumann) and, as a closely related variant, on compact Euclidean surfaces with several conic singularities, where repeated billiard reflections and iterated corner diffraction enter; and (B) the sharp threshold for scaling-critical magnetic Schrödinger operators $(-i\nabla-a(\hat x)/|x|)^2$ in $\mathbb R^n$, $n\ge3$, where the angular 1-form carries gauge-invariant structure that the planar case lacks.

Background

Bochner–Riesz means are smoothed spectral projections. For a nonnegative self-adjoint operator $L$ (throughout, a Laplace-type operator), the Bochner–Riesz means of order $\delta\ge0$ are $S^\delta_\lambda=(1-L/\lambda^2)_+^\delta$, where $t_+=\max(t,0)$; on $\mathbb R^n$ with $L=-\Delta$ this is the Fourier multiplier $(1-|\xi|^2/\lambda^2)_+^\delta$, which governs the summability of the inverse Fourier integral truncated to frequency disks. The central question, for each $1\le p\le\infty$, is the sharp summability index: the smallest exponent $\delta^*(p)$ such that $\|S^\delta_\lambda f\|_{L^p}\le C_p\|f\|_{L^p}$ holds with $C_p$ independent of the frequency parameter $\lambda$. On $\mathbb R^n$ the conjectured — and in the planar case proven — sharp value is $\delta_c(p,n)=\max(n\,|1/p-1/2|-1/2,\,0)$, and the same expression turns out to govern several related model operators.

For operators other than the flat Laplacian, sharp thresholds are known only for a short list of models: the Hermite oscillator (the harmonic-oscillator Hamiltonian $-\Delta+|x|^2$, whose eigenfunctions are Hermite functions; Lee–Ryu 2022), the twisted Laplacian (the phase space system associated with the planar harmonic oscillator; Jeong–Lee–Ryu 2025), and — the starting point of this problem — the two-dimensional flat cone (Jia–Zhang–Zheng 2025). A flat cone is the surface $X=(0,\infty)\times S^1_\sigma$ with metric $dr^2+\sigma^2r^2\,d\theta^2$: the plane with the origin replaced by a conical singularity of total cone angle $2\pi\sigma$. It is the local model for the singular points of a polygonal domain (each vertex is a cone point whose angle equals the interior angle) and of a compact Euclidean surface with conic singularities (a surface that is flat except at finitely many such points). Jia–Zhang–Zheng proved that on the flat cone, and on infinite planar sectors with Dirichlet or Neumann boundary conditions, $S^\delta_\lambda$ is $L^p$-bounded if and only if $\delta>\delta_c(p,2)$ for $p\ne2$ — the same threshold as the plane.

That result settles exactly one cone point. The surviving difficulty is global. On a bounded polygon, a wave emanating from one corner reflects specularly off the straight edges (the billiard dynamics) and diffracts at other corners — at a corner, a classical ray splits into a circular diffracted front — and those diffracted fronts are themselves reflected and re-diffracted, so the spectral measure accumulates contributions from infinitely many reflected and multiply diffracted wavefront pieces. Prior work on polygons controls single clusters (an $L^2$-normalized function with spectral support in a unit-width window $[\lambda,\lambda+1]$), not the full multiplier: Hassell–Hillairet–Marzuola (2009) proved eigenfunction concentration estimates for polygonal billiards (high-frequency eigenfunctions can concentrate along unstable periodic orbits and near corners), and Blair–Ford–Marzuola (2018) proved the sharp $L^2$-to-$L^p$ spectral cluster bounds on polygonal domains. Sogge (2002) obtained non-sharp Bochner–Riesz ranges on compact manifolds with boundary, and Guillarmou–Hassell–Sikora (2013) settled sharp restriction and spectral-multiplier questions on non-trapping asymptotically conic manifolds — a complete, boundary-free geometry where repeated reflection never occurs. Jia–Zhang–Zheng state that no sharp Bochner–Riesz result is known on any compact manifold with or without boundary except the flat torus, and identify polygonal domains as the intended next case.

The second strand concerns magnetic Schrödinger operators. For a magnetic vector potential $A$, the operator $H_A=(-i\nabla-A)^2$ describes a charged particle in a magnetic field; its scaling matches the spectral scaling $\lambda\mapsto\lambda^2$ only if $A$ is homogeneous of degree $-1$, and such potentials are called scaling-critical. The simplest planar example is the Aharonov–Bohm potential $\alpha\,d\theta$ (a magnetic flux $\alpha$ concentrated at the origin), for which Miao–Yan–Zhang (2024) proved the same if-and-only-if threshold $\delta>\delta_c(p,2)$ in $\mathbb R^2$, and stated that the critical-magnetic problem remains open in higher dimensions. In the plane, any smooth pure-magnetic critical potential is gauge-equivalent to this settled constant-flux case (two potentials differing by a gradient $\nabla\phi$ give unitarily equivalent operators), so the genuinely open generalization is dimension $n\ge3$, where the angular part $a$ of $A(x)=a(\hat x)/|x|$ (with $\hat x=x/|x|$) is a 1-form on the sphere $S^{n-1}$ whose gauge-invariant content cannot be reduced to a single flux number.

Problem Statement

Determine the sharp Bochner–Riesz summability threshold — an if-and-only-if criterion in $\delta$ and $p$ for uniform-in-$\lambda$ $L^p$ boundedness of $S^\delta_\lambda=(1-L/\lambda^2)_+^\delta$ — in the following two settings, which share one objective: extending the sharp threshold criterion beyond the settled single-singularity models, to operators whose spectral measures contain global diffractive structure (component A) or genuinely higher-dimensional scaling-critical magnetic structure (component B).

(A) Let $\Omega\subset\mathbb R^2$ be a bounded polygonal domain with finitely many vertices (interior angles in $(0,2\pi)$, reentrant angles allowed), and let $L$ be the Dirichlet or the Neumann Laplacian on $\Omega$. Determine, for each $1\le p\le\infty$ with $p\ne2$, the sharp exponent $\delta^*_A(p)$ such that $\|S^\delta_\lambda\|_{L^p\to L^p}\le C$ for all $\lambda>0$ if and only if $\delta>\delta^*_A(p)$; in particular, decide whether $\delta^*_A(p)$ equals the Euclidean and single-cone value $\delta_c(p,2)=\max\{0,\,2|1/2-1/p|-1/2\}$, and identify which geometric mechanisms (gliding or periodic billiard orbits, single versus iterated corner diffraction, coupling between distant corners) force the threshold when it differs. A closely related variant of the same objective is the Laplacian on a compact two-dimensional Euclidean surface with finitely many conic singularities and flat metric elsewhere; note that the doubling construction identifies a polygon with boundary conditions only with the special multi-cone surfaces carrying a reflective symmetry (cone angles twice the interior angles, Dirichlet/Neumann corresponding to the antisymmetric/symmetric summands), so the general multi-cone surface is a separate variant, not an equivalent formulation.

(B) In $\mathbb R^n$ with $n\ge3$, let $A(x)=a(\hat x)/|x|$ be a scaling-critical magnetic potential, where $\hat x=x/|x|$ and $a$ is a smooth real 1-form on the sphere $S^{n-1}$, and let $H_A=(-i\nabla-A)^2$, understood as the Friedrichs self-adjoint extension, following the convention of Miao–Yan–Zhang in the settled planar case. Determine, for each $1\le p\le\infty$ with $p\ne2$, the sharp exponent $\delta^*_B(p,n,a)$ such that $(1-H_A/\lambda^2)_+^\delta$ is uniformly $L^p(\mathbb R^n)$-bounded if and only if $\delta>\delta^*_B(p,n,a)$; in particular, decide whether $\delta^*_B(p,n,a)=\delta_c(p,n)=\max\{0,\,n|1/2-1/p|-1/2\}$ for every such $a$ (as it does for the planar constant-flux Aharonov–Bohm case in $n=2$), and if not, determine exactly which gauge-invariant data of $a$ (for instance the fluxes of $a$ over closed curves on $S^{n-1}$ and the spectrum of the associated angular operator $\mathcal L_A=(i\nabla_{S^{n-1}}+a)^2$) the threshold depends on.

Both components (A) and (B) belong to the question set out above.

References

1. Qiuye Jia, Junyong Zhang, Jiqiang Zheng, 'Bochner-Riesz means on a conical singular manifold', arXiv preprint, 2025. arXiv:2510.26059, https://arxiv.org/abs/2510.26059
2. Changxing Miao, Lixin Yan, Junyong Zhang, 'Bochner-Riesz means for critical magnetic Schrödinger operators in $\mathbb R^2$', arXiv preprint, 2024. arXiv:2405.02531, https://arxiv.org/abs/2405.02531
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