A sharp divide at p = 2
An arXiv preprint says the full horizontal Riesz transform has a uniform weak type (1,1) bound for real-valued functions: its constant is at most 2, regardless of horizontal dimension, homogeneous dimension, step or group structure. For real-valued inputs, the same analysis gives an Lp bound with a constant depending only on p when p is greater than 1 and at most 2. For p greater than 2, however, the authors conclude that no finite constant can make that bound hold for every stratified Lie group and every input in Lp. The result places p = 2 at a dividing line in the class studied: uniform control reaches that point, but not the entire higher-p range.
This is not a study with participants, measurements or an empirical data set. It is a proof-based analysis over the class of stratified Lie groups, with a constructed sequence of groups used for the high-p part rather than a participant sample.
A uniform endpoint estimate
Weak type (1,1) is the borderline estimate in the paper. In ordinary language, it controls how large the set of points can be where the transform becomes large, rather than promising a conventional L1 bound at the endpoint. For real-valued functions, the constant in that control is no more than 2 and stays independent of the group's principal parameters. The paper also extends the endpoint statement to complex-valued functions, with a constant of at most 4.
That uniformity is the notable feature: the estimate is not tied to one fixed horizontal dimension or one fixed step. It is stated across the stratified groups covered by the theorem, even as those parameters vary.
The proof turns to an obstacle problem
To reach the endpoint, the authors use a fractional obstacle problem adapted to stratified Lie groups. The approach relies on functional calculus, a framework for working with the sub-Laplacian, rather than on the Fourier transform. That is the route the paper uses for its group-uniform endpoint theorem.
At the core is a decomposition for a nonnegative input. When the fractional order is greater than 0 and at most 2, the input can be separated into a nonnegative remainder and a fractional sub-Laplacian term. The remainder is bounded above by a chosen level and preserves the input's L1 mass, meaning the total integrated size is unchanged. This mass-preserving split is one of the proof's central pieces.
The paper also identifies where this auxiliary construction is active. On the set where its obstacle-problem function is positive, the remainder equals the obstacle level apart from a measure-zero exception, and the size of that set is controlled by the input mass. When the fractional order is at least 1, the function's horizontal gradient vanishes outside that set. These localization properties help the proof keep track of the endpoint estimate.
Why higher p breaks the pattern
The low-p result is obtained through Marcinkiewicz interpolation, which converts the endpoint information into an Lp estimate for every p strictly above 1 and up to 2. Its constant depends only on p, not on the group parameters. The theorem therefore gives one finite, group-independent constant for each p in that interval.
For p greater than 2, the paper takes a different route. It constructs a sequence of stratified Lie groups whose horizontal dimension stays fixed at 5 while the number of steps increases without bound. Along that sequence, the Lp norms of the horizontal Riesz transforms diverge. This is an existence construction, not a claim that the norms diverge on every individual stratified Lie group.
The high-p construction modifies a Schrödinger-operator example, approximates its potential with polynomials through Mosco convergence of Dirichlet forms, and transfers the resulting behavior to stratified Lie groups.
That distinction is crucial. The high-p conclusion says that no finite universal constant can make the stated Lp inequality hold for every group and every input. It does not say that every group fails the estimate, nor does it settle what happens for special classes with structural or step restrictions.
A theorem, not a measurement
Nothing in the paper is a clinical, biological or human finding. The evidence is a proof-based analysis of operators on stratified Lie groups, with constructed sequences rather than participants or empirical observations. It therefore does not provide an observed effect in people or other biological systems.
The supplied document is an arXiv preprint identified as version 2, dated 4 September 2026. The supplied material does not report a journal or peer-review status, so the findings should be read as a mathematical preprint rather than as a settled published result.
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