A proof-based mathematical study reports uniform bounds for bilinear Bochner-Riesz means on the complex unit sphere, including a restricted-input case in which the required smoothness threshold falls from d+1 to d. The bound controls the output formed from two input functions by the product of their input norms, with a constant that stays independent of the spectral scale R when the stated exponent and smoothness conditions hold. The paper asks whether this two-input construction maps a product of input spaces into an output space under the standard Hölder relation, and what smoothness is sufficient.
The model is the complex unit sphere for n at least 2. Its ordinary topological dimension is 2n-1, while its homogeneous dimension is 2n. The relevant sub-Laplacian is formed from the sphere's Laplace-Beltrami operator and the square of the generator of its circle action.
To analyze the operator, the authors break functions into invariant harmonic pieces, each labeled by two harmonic degrees. On each piece, the sub-Laplacian acts by a fixed eigenvalue, while the circle-action generator acts by the imaginary unit times the difference between those degrees. This joint spectral decomposition lets the proof handle the two spectral variables together.
Thresholds vary with the exponents
The amount of smoothness required is not one fixed number. At the listed endpoint with input exponents 2 and 2 and output exponent 1, the threshold is 0. With inputs 2 and infinity and output 2, it is (d-1)/2, while the all-infinity endpoint has threshold d minus one-half. Two other listed endpoints, with inputs 1 and 1 producing output exponent one-half and inputs 1 and 2 producing two-thirds, require d+1 and (d+1)/2 respectively. With inputs 1 and infinity producing output exponent 1, the reported threshold is Q/2.
The proof follows the spectrum
One of the paper's main tools is an L1-to-L2 restriction-type estimate for the joint functional calculus, the framework used to handle the paired spectral variables. The authors describe these estimates as a way to reduce the required smoothness from the homogeneous dimension Q to the topological dimension d. The analysis also derives linear weighted Plancherel estimates with large powers of the weight and proves a bilinear weighted Plancherel estimate for the two-variable spectral multiplier kernel. To cover the wider exponent range, the proof establishes bounds at selected points, uses symmetry between the inputs, and applies bilinear interpolation.
Sharper results under restrictions
The clearest threshold improvement appears when the inputs are not arbitrary. A separate result places them in spectral classes called A_R and B_R, which impose specified restrictions on their spectral components. In Region V, the two inputs must also have the same parity. Under those added assumptions, the smoothness threshold moves from d+1 to d. That is a result for the restricted classes named in the theorem, not a general replacement for the full-input condition.
A related theorem studies near-diagonal localized inputs. It states that the Euclidean dimension appearing in the threshold is replaced by d-1+s, except in Region V. Here s is the localization parameter. At the endpoint where all three exponents are infinity, the proof requires the smoothness parameter to be greater than d+s-3/2.
The boundary of the claim
The qualifications are important. The full-input result is stated only for its listed admissible exponent regions, and the endpoint thresholds vary with the exponent triple. The restricted improvement applies only to A_R and B_R, while the localized result applies only to near-diagonal inputs. Because the stated conditions require smoothness above the thresholds, the supplied analysis does not settle what happens at equality. It also does not establish that the restricted or localized threshold statements extend to arbitrary full inputs.
The document is an arXiv preprint, version 2, dated 28 Aug 2026. Its subject is mathematical functions and spectral components on a sphere, so the conclusions are operator-boundedness statements under specified assumptions, not findings about human outcomes. The authors interpret the work as showing that a topological-dimensional phenomenon persists in bilinear theory across the admissible Banach and non-Banach regions.
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
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