{"id":"e6246d55-d150-4afa-a318-6ce7e67eebf9","arxiv_id":"2608.20267","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The full horizontal Riesz transform on any stratified Lie group satisfies a weak type (1,1) inequality with constant at most 2, uniformly in the dimension, step, and group structure.","lead":"A new proof shows that the horizontal Riesz transform on every stratified Lie group maps real L1 functions to weak-L1 with a universal constant at most 2, independent of dimension, step, and group structure. This is the first noncommutative analogue of a recent dimension-free Euclidean bound and extends endpoint singular-integral theory to sub-Riemannian settings.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the constant-2 weak-type theorem is supported; Proposition 2.1 contains a false but repairable range equality.","rationale":"The stress-test pass found no internal inconsistency that endangers Theorem 1.1. The obstacle method is coherent: coercivity via fractional Nash, existence and uniqueness of the minimizer, the Lewy-Stampacchia estimate giving 0 <= mu <= lambda, mass conservation via the cutoff lemma, complementarity, and the level-set argument together yield the constant 2. The false assertion Ran(L^{1/2}) = L^2 in Proposition 2.1 is a genuine error, but replacing it by density of the range repairs the proof; every later use of the Riesz transform only needs the isometry ||Rf||_2 = ||f||_2 on L^2. The two-sided Gaussian bound (2.7) is the least internally derived input, but it is standard and is not contested by the manuscript's own argument. Therefore the reader's CONDITIONAL verdict remains appropriate: conditional on the small correction, the theorem is sound.","tokens_in":13449,"tokens_out":37958,"duration_ms":301131,"concrete_test":"Consult VSC92 Theorem VIII.2.9 and confirm it gives the two-sided Gaussian bound (2.7) for all t > 0 on every stratified Lie group, with finite constants after using |B(x, sqrt(t))| comparable to t^{Q/2}. If the cited theorem has extra hypotheses (e.g., only large time or a narrower class of groups), the derivation of Lemma 2.2 and (2.11) must be reworked; this is the check that would settle the only substantive reliance on external input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing flaw in the central argument. Theorem 1.1 depends on (i) the L2 isometry of the Riesz transform, (ii) the obstacle decomposition with mass and complementarity, and (iii) the endpoint level-set argument. Each step checks. The imported two-sided Gaussian bound (2.7) is the least internally justified input, but it is a standard theorem for stratified groups (VSC92, Thm. VIII.2.9) and is only used to obtain finiteness, positivity, the Nash inequality (2.11), and the contraction lemma; none of these uses is circular. The one substantive error is in Proposition 2.1: Ran(L^{1/2}) is dense in, not equal to, (ker L^{1/2})^\\perp = L^2. The isometry U nonetheless extends uniquely to all of L^2 by density, and no later step (Theorem 1.1's (3.1)-(3.3), Lemma 2.5) invokes the false equality. The defect is genuine but not load-bearing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that on any stratified Lie group G, the full horizontal Riesz transform R = ∇_H L^{-1/2}, initially an L2 isometry, extends to real-valued L1(G) and satisfies the weak type (1,1) bound ||Rf||_{L^{1,∞}} ≤ 2||f||_1, with a constant independent of the horizontal dimension, homogeneous dimension, step, and group structure. The proof is variational: after splitting a real function into positive and negative parts, it constructs decompositions f± = μ± + L^{1/2}u± through a fractional obstacle problem (Theorem 1.2), with mass conservation, the bounds 0 ≤ μ± ≤ λ, and ∇_H u± supported on a set of measure at most λ^{-1}||f±||_1. The level-set estimate combines the measure of this support with Chebyshev's inequality applied to Rμ±, using the L2 isometry. The remainder of the paper establishes the obstacle decomposition for all 0 < α < 2 using heat-kernel Gaussian bounds, a fractional Nash inequality, spectral interpolation, and a Lewy–Stampacchia estimate. The Euclidean case G = R^m recovers the recent dimension-free theorem of Ouyang–Spector–Stockdale.","tokens_in":13653,"tokens_out":23304,"duration_ms":201492,"significance":"If correct, this is a substantial result: it gives a universal weak-type constant 2 for the full horizontal Riesz transform on every stratified group, including Heisenberg, Heisenberg-type, Engel, and free Carnot groups, with no dependence on any dimension or step. The proof is self-contained after standard external tools (spectral theory, Folland's subelliptic Sobolev spaces, and the VSC92 two-sided heat-kernel bounds), the constant emerges cleanly from an exact L2 isometry and a mass-preserving obstacle decomposition, and there is no circularity or free parameter. The paper is clearly written and the main line from the variational decomposition to the endpoint estimate is checkable. My reading found no load-bearing flaw; the mathematical inaccuracies are local and repairable.","major_comments":[],"minor_comments":[{"comment":"The displayed chain Ran(L^{1/2}) = (ker L^{1/2})^\\perp = L^2(G) is false for the unbounded self-adjoint operator L: its range is dense in (ker L)^\\perp, but it is not equal to it. This does not affect the conclusion, because the isometry U is defined on the dense subspace Ran(L^{1/2}) and therefore extends uniquely to L^2, but the proof should be rewritten using density rather than equality.","section":"Section 2.1, proof of Proposition 2.1"},{"comment":"For complex-valued v,w ∈ Vα the kernel representation should have a complex conjugate on the second factor, namely (v(x)-v(xz)) overline{(w(x)-w(xz))} Jα(z) dz dx; as written the identity is not the correct sesquilinear form. All later applications are to real functions, so this is a local correction rather than a substantive gap.","section":"Section 2.2, Eq. (2.4)"},{"comment":"In the Borel–Cantelli step, the displayed bound on the sum gives at most ∑_{r≥1} 2^{-r} = 1, so the strict inequality '< 1' is not literally correct; it should be '≤ 1' or, more simply, '< ∞'. The argument only needs finiteness of the sum.","section":"Section 3, proof of Theorem 1.1"},{"comment":"The two-sided Gaussian estimate (2.7) and the resulting Nash inequality (2.11) are imported from VSC92 with constants that may depend on the group. The paper should state this explicitly so that the universality claim is clearly understood as applying only to the final weak-type constant 2, not to the auxiliary constants used in the existence proof.","section":"Section 2.2, Eq. (2.7)"}],"recommendation":"minor_revision","confidential_remarks":"The central theorem is original and the proof is essentially sound; the only substantive defect is the repairable range equality in Proposition 2.1. I found no circularity, no hidden free parameter, and no novelty concern. The paper is a good fit for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nBottom line: this is the real thing. The theorem removes all group-dependent constants from the endpoint estimate and gives the clean constant 2 for every stratified group, which is exactly the kind of milestone result the area needs. The proof genuinely transfers the OSS26 obstacle method to the noncommutative setting, using the heat semigroup and Dirichlet form rather than Fourier analysis. The variational decomposition, the mass conservation argument, and the level-set splitting all check out; the constant 2 falls out naturally from splitting into the contact set and the Chebyshev contribution.\n\nThe one real error is Proposition 2.1. The claim that Ran(L^{1/2}) = (ker L^{1/2})^\\perp is false for an unbounded operator; only density holds. The fix is trivial: density alone lets U extend uniquely to L^2, and no later step uses the false equality. So this is a repairable defect, not a fatal one.\n\nMinor soft spots: the two-sided Gaussian heat kernel bound (VSC92, Thm. VIII.2.9) is doing essential work, but it is standard for stratified groups and is used only to get finiteness, positivity, and the Nash inequality. No circularity there. The L1 extension via Borel–Cantelli is standard. The AI-use declaration is transparent and does not affect the mathematics.\n\nThe citation pattern is honest: OSS26 is cited as the Euclidean antecedent and for recovery in the abelian case, and the Heisenberg-type literature is placed correctly. No free parameters, no invented entities.\n\nFor peer review: yes, send it out. Ask the referee to require the Prop 2.1 correction and a brief comment on the density argument; that's a minor revision, not a restart.\n\nBest,\n[Your name]","headline":"Strong, important result: uniform weak-type (1,1) bound with constant 2 for the full horizontal Riesz transform on every stratified Lie group; the only genuine defect, a false range equality in Prop 2.1, is easily repaired and not load-bearing.","tokens_in":14167,"tokens_out":9413,"would_cite":true,"duration_ms":76588,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["42B20","22E30","43A80","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"Theorem 1.1: the full horizontal Riesz transform on any stratified Lie group is weak type (1,1) with constant at most 2, uniformly in all group parameters.","keywords":["Riesz transform","weak type (1,1)","stratified Lie group","sub-Laplacian","fractional obstacle problem","heat semigroup","dimension-free estimates","Lewy-Stampacchia estimate"],"falsifier":"Compute on a concrete stratified group, for instance the Heisenberg group, the ratio $\\lambda |\\{x : |Rf(x)|>\\lambda\\}| / \\|f\\|_1$ for a sequence of $L^1$ functions approximating a point mass; if any limit exceeded $2$, Theorem 1.1 would be wrong. Alternatively, verify the proof's premise: if a stratified group's heat kernel failed the two-sided Gaussian estimate (2.7), the fractional Nash inequality would fail and the obstacle minimizer would not be guaranteed to exist.","tokens_in":13256,"feed_emoji":"🧮","tokens_out":8272,"duration_ms":67866,"temperature":0.7,"pith_summary":"The paper proves that on any stratified Lie group, the full horizontal Riesz transform $\\nabla_H \\mathcal{L}^{-1/2}$ sends real-valued $L^1$ functions to weak $L^1$ with constant at most $2$. The constant is independent of the horizontal dimension, the homogeneous dimension, the step, and which stratified group is chosen. This gives the same universal constant recently obtained in Euclidean space, extended to Heisenberg, Engel, and free Carnot groups, among others. A sympathetic reader would care because it shows the endpoint bound is a structural fact about the sub-Laplacian and its heat semigroup, not an artifact of Fourier analysis or Euclidean symmetry.","feed_headline":"One constant, 2, for Riesz transforms on every stratified group","feed_subtitle":"The horizontal Riesz transform maps L1 to weak L1 with universal bound 2, independent of dimension, step, or group.","key_machinery":"The load-bearing object is the fractional obstacle problem on the cone $K_\\alpha = \\{v \\in L^1 \\cap V_\\alpha : v \\ge 0\\}$ for the energy $J_\\lambda^\\alpha(v) = \\tfrac12 E_\\alpha(v) - \\int (f-\\lambda)v$, where $E_\\alpha(v) = \\|\\mathcal{L}^{\\alpha/4}v\\|_2^2$ is the fractional Dirichlet form. Its unique minimizer $u$ satisfies $f = \\mu + \\mathcal{L}^{\\alpha/2}u$ with $\\mu = \\lambda - \\eta$, where $\\eta$ is the Lewy-Stampacchia density $0 \\le \\eta \\le (\\lambda-f)_+$; the proof of that inequality is what turns the abstract variational inequality into the explicit complementarity relation $\\mu = \\lambda$ on $\\{u>0\\}$. The heat semigroup enters through the two-sided Gaussian kernel bound, which yields the fractional Nash inequality $\\|v\\|_2^2 \\le C_{G,\\alpha} \\|v\\|_1^{2\\alpha/(Q+\\alpha)} E_\\alpha(v)^{Q/(Q+\\alpha)}$ used to make the functional coercive. The final ingredient is the $L^2$ isometry identity for the full horizontal Riesz transform, which replaces the Fourier analytic cancellation of the Euclidean argument.","core_discovery":"The paper's central result is Theorem 1.1: the operator $R = \\nabla_H \\mathcal{L}^{-1/2}$, initially defined on $L^1 \\cap L^2$, extends uniquely to a continuous linear map from real-valued $L^1(G)$ to $L^{1,\\infty}(G;\\mathbb{R}^m)$, and $\\|Rf\\|_{L^{1,\\infty}} \\le 2\\|f\\|_1$ for every $f$. The proof is carried by an obstacle decomposition: every nonnegative $f \\in L^1 \\cap L^2$ is written as $f = \\mu + \\mathcal{L}^{\\alpha/2}u$ with $0 \\le \\mu \\le \\lambda$, total mass preserved, $\\mu = \\lambda$ on the set where $u > 0$, and $\\lambda |\\{u>0\\}| \\le \\|f\\|_1$; for $\\alpha \\ge 1$ the horizontal gradient $\\nabla_H u$ vanishes outside that set. The weak-type estimate follows by applying this at $\\alpha = 1$ to $f^+$ and $f^-$, then splitting $Rf = R\\mu + \\nabla_H u$. Chebyshev's inequality and the exact $L^2$ identity $\\|Rg\\|_2 = \\|g\\|_2$ bound the $\\mu$ contribution by $\\|f\\|_1$, while the measure bound on the support of $\\nabla_H u$ gives the matching contribution, summing to constant $2$.","pith_inferences":["Because the proof uses only the heat semigroup, Dirichlet form, and a Nash inequality, the same constant-$2$ mechanism is likely to extend to any noncommutative or infinite-dimensional setting whose sub-Laplacian has a heat kernel satisfying two-sided Gaussian bounds, such as complete Riemannian manifolds with doubling and Poincaré inequalities.","The exact mass conservation and complementarity suggest a 'sharp function' interpretation of $Rf$ via balayage; one could test whether the weak-type constant $2$ is asymptotically achieved by concentrating $f$ near the identity on groups with large step.","A testable extension would be to replace the full horizontal gradient by a single component and see whether the obstacle decomposition can be adapted, since the $L^2$ isometry for a single component fails.","The parameter $\\alpha$ in Theorem 1.2 is not used in the endpoint proof; pushing the same decomposition to $\\alpha > 1$ might yield weak-type estimates for higher-order Riesz-like operators built from $\\mathcal{L}^{-\\alpha/2}$."],"forward_implications":["Theorem 1.1 gives a weak $(1,1)$ bound with constant $2$ for each component $X_j\\mathcal{L}^{-1/2}$, since $|X_j\\mathcal{L}^{-1/2}f| \\le |Rf|$ componentwise.","Specializing to $G = \\mathbb{R}^m$, Theorem 1.1 recovers the dimension-free Euclidean result of [OSS26].","The result applies uniformly to every stratified group, including Heisenberg-type groups, Engel groups, and free Carnot groups of arbitrary rank and step, with no dependence on the homogeneous dimension or horizontal dimension.","For complex-valued functions, the same argument yields a weak-type bound with constant $4$; the paper makes no claim that $4$ is optimal.","Theorem 1.2 supplies a fractional balayage decomposition valid for every $0<\\alpha<2$, which may be of independent use in endpoint estimates for other fractional differential operators."],"supporting_citations":[{"why":"Supplies the Euclidean dimension-free weak-type theorem that this paper generalizes and the variational/obstacle method being adapted.","marker":"[OSS26]"},{"why":"Provides the two-sided Gaussian heat-kernel bounds on stratified groups that justify the kernel representation and the fractional Nash inequality.","marker":"[VSC92, Theorem VIII.2.9]"},{"why":"Gives the Schwartz smoothness and scaling of the heat kernel, the density of compactly supported functions in the fractional domain, and the subelliptic Sobolev space facts.","marker":"[Fol75]"},{"why":"Provides the background homogeneous-group theory showing that the components are Calderón–Zygmund operators and supplies the structural framework.","marker":"[FS20]"},{"why":"Introduced the decomposition by Dirac masses that the obstacle method extends.","marker":"[SS21]"},{"why":"Represents the positive distribution $\\eta$ as a Radon measure, giving the Lewy–Stampacchia density bound.","marker":"[LL01, Theorem 6.22]"},{"why":"Yields the horizontal truncation identity $\\nabla_H(v-c)_+ = 1_{\\{v>c\\}}\\nabla_H v$, used to show $\\nabla_H u$ vanishes off the set $\\{u>0\\}$.","marker":"[Gar16, Proposition 5.4]"}],"fun_headline_variants":["Riesz transforms get universal weak bound 2 on all stratified groups","Dimension-free weak L1 bound for stratified Riesz transforms: 2","One constant for all: Riesz transforms on stratified groups","Stratified groups: Riesz transforms have weak type (1,1) constant 2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the imported two-sided Gaussian bound for the heat kernel of the sub-Laplacian, Eq. (2.7): the bound must hold on every stratified group with finite constants, because it is what makes the fractional Nash inequality (2.11) hold and therefore guarantees that a minimizer $u$ exists.","fun_headline_variants_meta":{"raw":{"variants":["Riesz transforms get universal weak bound 2 on all stratified groups","Dimension-free weak L1 bound for stratified Riesz transforms: 2","One constant for all: Riesz transforms on stratified groups","Stratified groups: Riesz transforms have weak type (1,1) constant 2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001219,"raw_usage":{"total_tokens":5061,"prompt_tokens":1040,"completion_tokens":4021,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":656,"completion_tokens_details":{"reasoning_tokens":3950}},"tokens_in":656,"tokens_out":4021,"duration_ms":24560,"temperature":1.0,"reasoning_tokens":3950,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T19:21:39.380411+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute on a concrete stratified group, for instance the Heisenberg group, the ratio $\\lambda |\\{x : |Rf(x)|>\\lambda\\}| / \\|f\\|_1$ for a sequence of $L^1$ functions approximating a point mass; if any limit exceeded $2$, Theorem 1.1 would be wrong. Alternatively, verify the proof's premise: if a stratified group's heat kernel failed the two-sided Gaussian estimate (2.7), the fractional Nash inequality would fail and the obstacle minimizer would not be guaranteed to exist.","supporting_citations":[],"review_version":1}