There is a particular satisfaction in finishing a paper that builds something from scratch. Not a refinement, not a generalization in the usual sense, but an actual construction, objects that didn’t exist before, assembled carefully from the ground up. Our recent work on Haar bases for multi-parameter twisted structures gave me that feeling, and it also reminded me how much foundational infrastructure the rest of harmonic analysis takes for granted.
The motivation comes from geometry. The Cauchy-Szegő projection on a broad class of Siegel domains, and the quotient structures of nilpotent Lie groups observed by Nagel, Ricci, and Stein, give rise to a class of twisted multi-parameter singular integrals that fall entirely outside the scope of existing product and flag theories. The obstruction is fundamental: the class of standard product operators is not closed under passage to a quotient subgroup. When you project a well-behaved product kernel down to a quotient space, the independent singularities couple, producing a twisted kernel singular along intersecting hyperplanes. Continuous real-variable methods for these operators, tube maximal functions, Littlewood-Paley area functions, atomic decompositions for twisted Hardy spaces, have been developed in other work by some of my collaborators. What this paper does is build the discrete side: the dyadic martingale theory and adapted Haar wavelet bases that the setting requires and that, until now, didn’t exist.
So we built it.
The core difficulty is that the geometry itself resists the standard toolkit. In the Euclidean setting, the quotient map π(x₁, x₂, x₃) = (x₁ + x₃, x₂ + x₃) from ℝ³ᵐ to ℝ²ᵐ induces a twisted tube geometry that organizes naturally into three dyadic systems: the standard product rectangles I × J, and two families of slanted tubes I ×ₜ J and I ᵗ× J tilted by the coupling. The key insight is that these slanted systems are not mysterious, they are exact pullbacks of the standard product system under two measure-preserving affine shears, T₂(x₁, x₂) = (x₁ − x₂, x₂) and T₃(x₁, x₂) = (x₁, x₂ − x₁). Once you see that, the Haar construction writes itself. Pulling back the standard tensor-product Haar basis through T₂ and T₃ gives two new systems B₂ and B₃, each a complete orthonormal basis for L²(ℝ²ᵐ), each supported on its corresponding slanted tubes, and each carrying cancellation in the natural rectified coordinates. The union of all three is a tight frame.
Corollary (Twisted Haar Frame). The union B₁ ∪ B₂ ∪ B₃ forms a tight frame for L²(ℝ²ᵐ) with frame bound 3. For any f ∈ L²(ℝ²ᵐ),
with unconditional convergence in L².
What makes this work cleanly is unitarity. Because T₂ and T₃ are measure-preserving, their pullback operators U₂ and U₃ are unitary on L², and this unitarity transports all the Lᵖ theory essentially for free. The twisted conditional expectations satisfy 
Theorem (Twisted BDG Inequalities). For any 1 < p < ∞ and i ∈ {1, 2, 3}, the twisted martingale square functions satisfy
with constants depending only on p and the dimension m.
You don’t reprove these estimates, you transport them. The hard work is identifying the correct shears and setting up the framework so that the conjugation is clean. The dyadic Littlewood-Paley equivalences follow by the same mechanism, using the pointwise identity 
The nilpotent setting is where the construction required the most care. On the quotient group N, the Shilov boundary of certain fundamental Siegel domains, a nilpotent Lie group of step two modeled as the quotient of the lifting group Ñ = H₁ × H₂ × H₃, standard dyadic grids simply don’t exist. Heisenberg groups don’t come equipped with them. In their place, one works with fractal tilings due to Strichartz and Tyson: unlike Euclidean dyadic cubes, these tiles partition the Heisenberg group at each scale and remain comparable to metric balls, but they are not nested across scales, the clean inclusion property that makes Euclidean martingale theory work is absent from the outset. We project products of these fractal tiles from Ñ down to N to define what we call raw projected shards. These capture the correct quotient geometry and remain comparable to the quotient tubes from the continuous theory. But the failure of nesting means they are not the right objects for an exact Haar theory, and in one of the three geometric regimes the diagonal union structure prevents a clean rectification as well.
This distinction between raw projected shards and analytic dyadic shards is one of the things I find most interesting about the paper. The raw shards tell you what the geometry wants to be. The analytic shards are what you actually build the analysis on, exact inverse images of product blocks under measure-preserving rectifications, nested across scales, and uniformly comparable to the raw shards in all relevant regimes. More precisely, the admissible quotient dyadic structure on N is a union R = R^(1) ∪ R^(2) ∪ R^(3), where R^(1) and R^(2) are exact product families in rectified coordinates, and R^(3) is defined by pulling back R^(2) under the central shear Θ(z, t₁, t₂) = (z, t₁ − t₂, t₂). The three systems are nested across scales, and each analytic shard R satisfies a tube comparison
for a uniform integer σ depending only on the dimensions, so the analytic dyadic structure remains geometrically faithful to the quotient tube geometry from the continuous theory.
The twisted Haar systems on N are then defined by the same unitary pullback strategy. For k = 1, 2, the Haar functions are standard tensor-product Haar bases on the product filtrations in rectified coordinates. For k = 3, they are defined by composition with Θ, giving a third complete orthonormal basis B^(3) supported on the analytic dyadic shards of type III. The main results on N run exactly parallel to the Euclidean case:
Theorem (Twisted Nilpotent Haar Tight Frame). The union is a tight frame for L²(N) with frame bound 3, and
with unconditional convergence in L²(N).
Theorem (Dyadic Littlewood-Paley Estimates on N). For every 1 < p < ∞ and k ∈ {1, 2, 3},
The proof strategy is the same: unitarity of the pullback U₃ = · ∘ Θ reduces the k = 3 estimates to the k = 2 case, which is the standard product Littlewood-Paley theorem in rectified coordinates. The geometry does the heavy lifting once the correct rectification is identified.
The paper contains no applications in the usual sense, and I think it’s worth being direct about that. This is infrastructure. The twisted Haar systems and dyadic shards are meant to serve as the discrete foundation for whatever comes next in this setting, paraproducts, T(1) theorems, commutator estimates, in the same way that classical dyadic theory undergirds the continuous theory in product settings. Doing so forced choices that have no classical analogue, particularly the separation of the geometric objects (raw shards) from the analytic objects (analytic shards). That distinction wasn’t obvious at the outset; it emerged from working through what an exact Haar theory actually requires, and from sustained collaborative argument about where the geometry is flexible and where it genuinely resists.
Open problems remain, as they always do. The most natural next step is extending this framework to continuous operators, showing that paraproducts and Calderón-Zygmund operators in the twisted setting can be controlled using this discrete theory. That passage is non-trivial: on the continuous side, the twisted kernel estimates involve cancellation conditions along intersecting hyperplanes that don’t decompose cleanly into the three rectified systems, and reconciling those estimates with the discrete cancellation structure established here will require genuinely new ideas. The discrete-to-continuous bridge is never automatic, and in the twisted quotient setting the coupling of singularities makes it harder than usual. But the foundation is there now, and that’s what this paper was for.
The paper is on arXiv. I’m grateful to my collaborators for their patience, precision, and willingness to argue carefully about the things that mattered.