Boundary Sparse Bounds for Planar Bochner–Riesz Means

Definitions and mathematical statement

Let $0<\lambda<\tfrac12$. In the notation of the supplied record, $R_t^\lambda$ denotes the planar Bochner--Riesz means, $\operatorname{Sp}(p,q)$ denotes the sparse-bound property under consideration, and $T_2(\lambda)$ denotes the necessary-condition trapezoid in the plane of reciprocal exponents. Write $S_\lambda=P_2P_3$ for its designated boundary segment.

The source-relative question is whether, for every $\lambda\in(0,\tfrac12)$ and every admissible exponent pair $(p,q)$, \[ (1/p,1/q)\in S_\lambda \quad\Longrightarrow\quad R_t^\lambda\text{ satisfies }\operatorname{Sp}(p,q). \] Here "admissible" refers to the exponent domain intended by the report; that domain is not specified in the supplied record.

This formulation retains unresolved source notation. The supplied material gives neither the multiplier formula and parameter range for $R_t^\lambda$, nor the defining inequality for $\operatorname{Sp}(p,q)$, nor the coordinates of $P_2,P_3$ or defining inequalities of $T_2(\lambda)$. It also does not specify whether the sparse bound is uniform in $t$, whether a maximal operator is involved, or whether both endpoints of $P_2P_3$ are included. These details must be recovered from the cited report before the question becomes a self-contained mathematical statement.

As background, the supplied report excerpt asserts that the sparse bound is obtained for reciprocal-exponent pairs in $T_2(\lambda)\setminus P_2P_3$, and describes the behavior on $P_2P_3$ as open. This is retained as a supplied source claim, not an independently verified theorem or current literature assessment.

Short English statement

For every $0<\lambda<\tfrac12$, determine whether planar Bochner--Riesz means satisfy all sparse bounds corresponding to the designated boundary segment $P_2P_3$. The supplied record omits the definitions needed to specify the segment, sparse inequality, exponent domain, and scale quantifiers precisely.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005553$, source problem code catalogue entry, titled "Boundary Sparse Bounds for Planar Bochner--Riesz Means." The record belongs to the Oberwolfach Reports Open Problems collection and records the proposed year as 2023; no individual proposer is supplied.
• Supplied report citation: Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory (2023), Oberwolfach Reports, pp. 1397--1452, DOI 10.4171/owr/2023/25. https://doi.org/10.4171/owr/2023/25.
• Supplied literature source, labelled "OWR 2023 25": https://publications.mfo.de/bitstream/handle/mfo/4055/OWR_2023_25.pdf?sequence=1.
• Literature assessment date: 2026-08-22. The supplied dated curation labels the problem open, states that the report explicitly leaves the planar boundary-segment sparse bound open, and says that no later proof was found. This assessment is supplied dated curation and has not been independently verified here.
• The registry labels the statement corrected_verified and says it was checked against the official report and preserves its hypotheses, notation, and mathematical target. That verification is a supplied provenance assertion; the report has not been independently consulted here.