An arXiv preprint reports a universal bound for horizontal Riesz transforms on stratified Lie groups. For real-valued functions in L1(G), the full transform extends continuously to weak L1(G; R^m), with a weak-type constant of at most 2. In plain language, the result controls the size of the set where the transform becomes large, based on the total size of the original function.
The bound is uniform: the constant 2 does not depend on the horizontal dimension, the homogeneous dimension, the step of the group, or the group structure. The same numerical ceiling is therefore asserted across the entire class of stratified Lie groups covered by the theorem.
The document is an arXiv version 1 preprint dated 20 Aug 2026. It is a proof-based study, not a report from an empirical sample. Its objects are stratified Lie groups, their sub-Laplacians and Haar measure, and functions in specified mathematical domains.
What the bound means
The label weak type (1,1) describes an endpoint estimate. The input is measured in L1, the space used for functions with finite integrated absolute size, while the output is controlled through the size of the regions where it exceeds a chosen level. The theorem thus gives a threshold-based control of large outputs rather than an ordinary point-by-point bound.
The statement applies to the full vector of horizontal component transforms, not just to one component. That vector takes real-valued inputs into weak L1(G; R^m), so the result concerns the combined horizontal transform in the group setting.
The paper also establishes an exact summed squared-norm identity for the component transforms on L2(G). In practical mathematical terms, combining the component outputs through their squared norms gives an exact relation to the corresponding L2 size of the input.
The setting is noncommutative: the analysis is carried out on a stratified Lie group rather than on an empirical population or a conventional data table. The supplied analysis contains no participants, treatment groups, numerical experiment, or statistical comparison.
A proof built around an obstacle
The proof uses a fractional obstacle problem built from the heat semigroup and its associated Dirichlet form, adapted to the stratified structure. Instead of estimating the endpoint transform in one step, the argument first breaks a nonnegative input into a bounded term and a fractional operator term.
For a nonnegative f in L1(G)∩L2(G), a positive level λ, and a fractional range 0<α<2, the paper constructs nonnegative functions μ and u satisfying f=μ+L^(α/2)u in L2(G). The bounded term μ belongs to L1(G)∩L∞(G), while u belongs to L1(G) and the fractional domain D(L^(α/2)).
The decomposition has several properties that make the endpoint estimate possible. Almost everywhere, μ lies between min{f,λ} and λ. It preserves the L1 mass of f, and it equals λ on the positivity set Ω={u>0}. The paper also proves that λ|Ω| is no larger than the L1 norm of f, tying the threshold to the measure of that set.
When α is at least 1, u lies in the horizontal Sobolev domain and its horizontal gradient vanishes almost everywhere outside Ω. The gradient is therefore localized to a set whose measure is already controlled by the mass estimate.
How the constant is assembled
The endpoint proof attributes the final constant 2 to two contributions. Exact mass conservation and localization of the horizontal gradient control one part of the estimate. The exact squared-norm identity, together with Chebyshev’s inequality, controls the other. Combined, those contributions produce the stated weak-type constant at most 2.
The general real-valued L1 extension is then constructed by approximation from real-valued functions in L1∩L2. This carries the estimate from the function class used in the proof to the full real-valued L1 domain named in the main result.
The paper states a corresponding result for complex-valued inputs, but with a weak-L1 constant of at most 4 rather than 2. It explicitly makes no claim that the complex-valued constant is optimal, so the statement is a bound rather than a declaration of the best possible number.
The questions the proof leaves open
This work does not measure an outcome in people or in an experimental sample. It reports no empirical dataset, numerical experiment, statistical uncertainty, confidence interval, or finite-sample generalization. Its evidence is the analytic theorem and the mathematical arguments used to prove it.
The conclusion is limited to the stratified Lie groups covered by the paper. The supplied analysis does not establish the same universal bound for a broader class of groups, even though the constant is independent of several dimensions and structural parameters within the stated class.
The real-valued result is stated with a constant at most 2, which does not by itself establish that 2 is optimal. The complex-valued extension is stated with a constant at most 4, and the paper leaves open whether that number can be improved.
The finding is most relevant to analysts studying endpoint bounds and Riesz transforms on noncommutative groups. It is not a clinical, behavioral, or population-health finding, and it offers no basis for individual medical advice.
Disclosures in the preprint
The acknowledgements report partial support for S.-C. Mao from the China Postdoctoral Science Foundation, grant 2026M793367, and partial support for Y. Wang from the China Scholarship Council. Y. Zhang reports funding from the European Research Council under the European Union Horizon 2020 programme through grant agreement GEOSUB, No. 945655.
The authors state that OpenAI GPT-5.6 was used for brainstorming, preliminary ideas, and English-language editing, while the mathematical results and proofs were independently developed, checked, and verified by the authors.
Paper data and sources
Original title: Uniform weak type $(1,1)$ bounds for Riesz transforms on stratified Lie groups
Authors: Sheng-Chen Mao, Yaojun Wang, Ye Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text