An arXiv preprint reports a uniform boundedness theorem for bilinear Bochner–Riesz means tied to the sub-Laplacian on the complex sphere. The main result applies in listed admissible exponent regions when the smoothness parameter α exceeds the relevant threshold; the bound uses a constant independent of R. Put simply, one threshold for each allowed exponent region is enough to keep the two-function operation controlled across the theorem’s range of R. Here, a smoothness threshold is the lower value of α that must be exceeded for the result to apply.
Two dimensions, one operator
This is not an experiment or a dataset study. The mathematical inputs are functions f and g on the complex unit sphere, analyzed through spectral components of the sub-Laplacian rather than sampled participants or observations. For n ≥ 2, the sphere has topological dimension d = 2n − 1 and homogeneous dimension Q = 2n. The paper’s central question is how the sphere’s dimension enters the smoothness needed for boundedness.
How the bounds are built
The proof first establishes bounds at selected endpoint combinations of input and output exponents, then extends the allowed range through symmetry and bilinear interpolation. A restriction-type estimate for the joint functional calculus is identified as a main tool behind the restricted-input improvement, lowering the required smoothness from Q to d. The paper also adds weighted Plancherel estimates: one extends control to every nonnegative integer power N of the weight, and another covers multipliers supported on [0, N]^2 when 0 ≤ α1, α2 < 1/2.
The threshold map
At the named endpoints, the paper gives a detailed threshold map. For (p1, p2, p) = (2, 2, 1), the threshold is 0; for (2, ∞, 2), it is (d − 1)/2; and for (∞, ∞, ∞), it is d − 1/2. At (1, 1, 1/2), it is d + 1, while at (1, 2, 2/3), it is (d + 1)/2. The (1, ∞, 1) endpoint uses Q/2. The paper also reports losses of 3/2 and 1/2 orders relative to the cited Euclidean estimates at these endpoints.
The gain comes with conditions
The clearest improvement appears at the (1, 1) endpoint, but only after the input class is narrowed. With additional assumptions on the input functions, the sufficient threshold drops from d + 1 to d. The paper connects that gain to the restriction-type estimate, which lowers the required smoothness from the homogeneous dimension Q to the topological dimension d in this restricted setting. Because the statement depends on those added assumptions, it does not give the same threshold for unrestricted Lp inputs.
Localization shifts the thresholds
A further result treats a spectrally localized setting, where the analysis is limited to selected spectral components. There, the authors describe the threshold pattern as an analogue of the Euclidean one with n replaced by d − 1 + s, except in Region V. At the localized (1, 1) input-exponent endpoint, the sufficient condition is α > d − 1 + s. At the localized (∞, ∞, ∞) endpoint, it is α > d + s − 3/2.
The boundaries of the result
Taken together, the results support the authors’ view that a topological-dimensional phenomenon persists across several bilinear Banach and non-Banach regions. The coverage, however, is bounded by the exponent regions and strict inequalities stated in the results. The supplied analysis does not establish that the thresholds are necessary or sharp, whether equality cases hold, or whether the restricted and localized improvements extend to unrestricted inputs or beyond the listed regions. The work therefore maps a set of sufficient bounds rather than settling the full bilinear theory.
Where the work stands
The document is an arXiv v1 preprint dated 26 August 2026. Its acknowledgments name FIST grant SR/FST/MS-II/2019/51, ANRF project ANRF/ARG/2025/003732/MS, PMRF and institute post-doctoral fellowship support, and Louisiana EPSCoR, NSF, and Board of Regents Support Fund award OIA-2437963.
Paper data and sources
Original title: Bilinear Bochner--Riesz Means on the Complex Sphere
Authors: S. Bagchi, Md N. Molla, J. Singh, M. N. Vempati
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text