{"id":"a0f3f323-b703-4680-9cb0-d25764d10598","arxiv_id":"1908.07889","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors introduce α-Hermite BV spaces, perimeters, and capacities, and prove that they satisfy coarea, Sobolev, isoperimetric, trace, and isocapacity inequalities, together with an L1 mean-curvature result for a restricted perimeter.","lead":"This paper builds a bounded-variation, perimeter, and capacity theory for the Schrödinger-type operator H_α = Δ − (α−1)|x|^α on R^d. It proves Sobolev, isoperimetric, trace, and isocapacity inequalities, plus a mean-curvature theorem for a restricted version of the α-Hermite perimeter.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection to the central Sobolev claim: a one-line classical-variation argument proves Theorem 1.11(i) and bypasses the disputed Lipschitz step.","rationale":"The reader correctly identifies a real flaw in the proof of Theorem 1.5: for α∈[1,2) the function ψ(x)=x_k|x|^{(α-2)/2} is not Lipschitz, since its gradient blows up like |x|^{α/2-1} at the origin, and the J2 estimate in that proof relies on a Lipschitz bound. This is a legitimate proof gap in the manuscript as written. However, the strongest claim singled out by the reader, Theorem 1.11(i), does not actually depend on that approximation theorem: the α-Hermite variation dominates the classical total variation through the simple admissible choice φ=(ψ/2,ψ/2), whose zero-order contributions cancel. This makes every BV_Hα function a classical BV function, and the classical GNS inequality applies directly. The Sobolev inequality is therefore secure even if Theorem 1.5's proof needs repair. The reader's other concerns—the garbled indicator approximation in the coarea proof, the trace-theorem chain, and the α=1 perimeter constant—are real but localized issues that support a conditional verdict rather than acceptance. Since the central Sobolev claim is not undermined, the reader's CONDITIONAL verdict should remain unchanged.","tokens_in":29471,"tokens_out":27954,"duration_ms":286870,"concrete_test":"For arbitrary f∈BV_Hα(R^d) and ψ∈C_c^1(R^d;R^d) with |ψ|≤1, set φ=(ψ/2,ψ/2). Verify explicitly that φ is admissible, that div_Hαφ=divψ, and therefore that |∫ f divψ|≤‖∇Hαf‖(R^d); conclude |Df|(R^d)≤‖∇Hαf‖(R^d). Then apply the classical BV Sobolev inequality to prove Theorem 1.11(i) without relying on Theorem 1.5 or the coarea formula. If this two-step verification succeeds, the reader's Lipschitz concern does not threaten the paper's strongest claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified to the paper's strongest claim, Theorem 1.11(i). The flagged failure of ψ(x)=x_k|x|^{(α-2)/2} to be Lipschitz for α∈[1,2) does invalidate a step in the proof of Theorem 1.5, but that theorem is not the only route to the Sobolev inequality. For every f∈BV_Hα(R^d) and every test vector ψ∈C_c^1(R^d;R^d) with |ψ|≤1, take φ=(ψ/2,ψ/2)∈C_c^1(R^d;R^{2d}). Then |φ|²=|ψ|²/2≤1, so φ is admissible, and div_Hαφ=divψ exactly because the zero-order terms cancel. Hence |∫ f divψ dx|≤‖∇Hαf‖(R^d), meaning the classical distributional derivative Df is a finite vector measure with |Df|(R^d)≤‖∇Hαf‖(R^d). The classical Gagliardo-Nirenberg-Sobolev inequality for BV then gives Theorem 1.11(i) immediately, without invoking Theorem 1.5 or the coarea formula. Thus the central Sobolev claim stands on independent grounds within the paper's own definitions; the Lipschitz concern is a genuine proof gap but not a load-bearing threat to the strongest claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces spaces of functions of α-Hermite bounded variation, BV_Hα(R^d), associated with the operator H_α=Δ−(α−1)|x|^α for α∈[1,∞), using a 2d-component generalized gradient. It develops basic Banach-space properties, a lower semicontinuity result, an approximation theorem, a coarea formula, and Sobolev and isoperimetric inequalities for these spaces. It then defines an α-HBV capacity, establishes measure-theoretic and duality properties, proves trace and isocapacity equivalences, and introduces a restricted α-Hermite perimeter that restores complement symmetry. This restricted perimeter is used to extend the Barozzi–Gonzalez–Tamanini mean-curvature theorem to the α-Hermite setting.","tokens_in":29679,"tokens_out":16342,"duration_ms":247236,"significance":"If the gaps noted below are repaired, the paper would provide a coherent BV theory for a family of Schrödinger-type operators with polynomial potentials and would connect functional capacity, perimeter, and mean curvature in a nontrivial way. A genuine strength is that the central Sobolev inequality, Theorem 1.11(i), is sound: for any test vector ψ∈C_c^1(R^d;R^d), taking φ=(ψ/2,ψ/2) gives |∫ f divψ dx|≤‖∇_{H_α}f‖, so the classical distributional derivative is controlled and the classical Gagliardo–Nirenberg–Sobolev inequality applies. The introduction of the restricted perimeter is a useful device because PHα lacks the classical complement symmetry, and the ball estimates in Corollary 1.13 quantify this asymmetry explicitly. The main weaknesses are concentrated in the proofs of the approximation theorem, the coarea formula, the trace theorem, and one isocapacity direction; these are repairable but are not merely cosmetic.","major_comments":[{"comment":"The estimate for J2 in the proof of Theorem 1.5 asserts that ψ(x)=x_k|x|^{(α−2)/2} is Lipschitz on Ω for all α∈[1,∞). This is false for 1≤α<2: near x=0 one has |ψ(x)|≈|x|^{α/2} and |∇ψ(x)|≈|x|^{α/2−1}, which is unbounded when α<2. Since Theorem 1.5 is invoked in Theorem 1.6, Theorem 1.10, Theorem 1.11, Lemma 2.7, and Theorem 2.9, this is a load-bearing gap in the manuscript as written. The Sobolev inequality itself survives, because the test-vector argument with φ=(ψ/2,ψ/2) reduces Theorem 1.11(i) to the classical BV Sobolev inequality, but the approximation theorem still needs a separate proof for α∈[1,2), for instance using the Hölder continuity of ψ.","section":"Theorem 1.5, proof of the J2 estimate"},{"comment":"The reverse inequality in the coarea formula is not completed as written. The estimate is first obtained for smooth functions, and the passage to f∈BV_Hα uses Theorem 1.5 together with lower semicontinuity, but the displayed bound contains the term √2(α−1)∫∫_{f≥t}|x|^{α/2}dxdt. To pass this term through an approximating sequence f_k one needs convergence of ∫|f_k||x|^{α/2}dx, which is not part of the L1 plus variation approximation supplied by Theorem 1.5. This matters because the coarea formula is used in Theorem 2.2, Lemma 2.7, and Theorem 2.10. The missing inequality is likely true for α>1 via the test pair φ=(ψ,−ψ), which controls the weighted L1 norm by ‖∇_{H_α}f‖, but the proof must make this explicit.","section":"Theorem 1.10, proof after (1.12)"},{"comment":"The displayed chain of estimates in this part is not a valid derivation of the trace inequality. The layer-cake formula contains t^{p−1}dt, not t^{α−1}dt, and the passage from (∫_0^∞ μ({|f|>t})t^{p−1}dt)^{1/p} to ∫_0^∞ μ({|f|>t})^{1/p}dt is asserted without a supporting Hardy-type argument. As printed this step is the core of the implication and must be corrected or replaced by a standard trace-theorem argument for BV-type capacities.","section":"Theorem 2.9, proof of (iii)⇒(i)"},{"comment":"The assertion P(E_δ)→0 as δ→0 is false for a d-dimensional compact set M with nonempty smooth boundary. For M=closed unit ball, E_δ={1<|x|<1+r} has classical perimeter (d−1)ω_d[(1+r)^{d−1}+1], which tends to 2(d−1)ω_d, not 0. Consequently the claimed convergence ‖∇_{H_α}f_δ‖_{L1}→PHα(M) does not follow from the displayed inequalities, and the proof of (2.3)⇒(2.4) is incomplete. The desired limit may still be true, but it requires a sharper estimate of PHα({dist(·,M)<r})−PHα(M) rather than a bound by the perimeter of the shell.","section":"Theorem 2.10(ii), proof around E_δ"}],"minor_comments":[{"comment":"Several mathematical symbols are corrupted in the production text, including '/greaterorsimilar', '/nequal', and the spacing in 'V ariation'; these should be corrected in the final version.","section":"Global typesetting"},{"comment":"The notation PHα(B(0,s)^c) is used before the asymmetry of the perimeter is discussed; a forward reference to Remark 1.14 would help the reader.","section":"Corollary 1.13 and Remark 1.14"},{"comment":"The factor √2 in the comparison |∇f|≤|∇_{H_α}f|≤√2(|∇f|+√(α−1)|x|^{α/2}|f|) is consistent with the 2d-component definition of ∇_{H_α}, but this normalization is not explained and should be stated explicitly.","section":"Equation (1.11)"},{"comment":"The phrase that cap(·,BV(R^d)) is 'not only an outer measure' is slightly misleading: the proof establishes monotonicity, countable subadditivity, and Choquet-type continuity for compact and increasing families, not the usual outer-measure regularity; rephrasing would avoid confusion.","section":"Theorem 2.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a coherent framework and the central Sobolev inequality is independently verifiable, so I would not recommend rejection. However, the number of incomplete proofs in the approximation, coarea, trace, and isocapacity sections is substantial, and several are load-bearing. I would ask the authors to repair or replace the proofs of Theorem 1.5, Theorem 1.10, Theorem 2.9(iii)⇒(i), and Theorem 2.10(ii) before publication. The relationship to the Gaussian BV capacity program of Xiao and the Grushin-plane results of Liu should also be made more explicit in Sections 2.2–2.3."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper really does extend the Xiao/Liu BV-capacity-perimeter machine to the α-Hermite operators H_α = Δ − (α−1)|x|^α, and the central Sobolev inequality (Theorem 1.11(i)) is solid: as a quick stress test shows, for any admissible ψ, φ=(ψ/2,ψ/2) gives |∫ f divψ| ≤ ‖∇_{H_α}f‖, so classical BV-GNS applies directly. Second, the manuscript has a few genuine local gaps, most notably a false Lipschitz claim in the proof of Theorem 1.5. Neither gap kills the paper, but a referee should push for repairs.\n\nThe genuinely new items are the α-Hermite BV space, the complement asymmetry observation (1.15) with the restricted perimeter in Definition 1.15, and the mean-curvature theorem for the restricted functional. Those look new relative to Gaussian and Grushin settings. The main proofs follow the standard templates—coarea, capacity duality, trace—and the capacity inequalities are structurally sound. The counterexample P_{H_α}(B(0,r)^c)=∞ versus finite P_{H_α}(B(0,r)) is a nice touch, and the restricted perimeter is a natural fix.\n\nSoft spots, in increasing order of trouble:\n\n- Theorem 1.5: ψ(x)=x_k|x|^{(α−2)/2} is not Lipschitz for α∈[1,2); it is Hölder with exponent α/2. The J2 estimate needs a Hölder estimate rather than Lip(ψ). Easy repair, but as written the proof is invalid in that range.\n- The trace theorem chain in (iii)⇒(i) of Theorem 2.9 has a garbled displayed inequality (a stray t^{α−1} appears). Probably just a typo, but it should be rewritten.\n- Theorem 3.1 has a step about u vanishing outside E and measure-zero/inclusion issues that need to be made precise.\n- The analogues for ~P_{H_α} are asserted without proof; some are fine to omit, but if they are used later they should be stated or referenced.\n- Minor: the factor 2 in the α=1 perimeter relative to classical perimeter is consistent, but constants in Theorem 2.2 and 2.10 should be checked.\n\nNone of these is load-bearing for the main Sobolev/isoperimetric story. The paper is for researchers in geometric measure theory and harmonic analysis who want a BV-capacity toolkit for Hermite-type operators. It deserves a serious referee; I would not desk-reject. Send it to peer review and ask for a revision that fixes Theorem 1.5, the trace chain, and the mean-curvature details.","headline":"Solid extension of the BV-capacity-perimeter framework to α-Hermite operators, with a robust central Sobolev inequality, but the manuscript ships with a few fixable proof gaps.","tokens_in":30368,"tokens_out":4713,"would_cite":true,"duration_ms":69965,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B35","47A60","32U20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the space of functions of bounded variation associated with the α-Hermite hydrogen-atom operator satisfies Sobolev, isoperimetric, and isocapacity inequalities, and that a restricted perimeter repairs the failure of…","keywords":["α-Hermite bounded variation","Hermite operator","perimeter","capacity","Sobolev inequality","isoperimetric inequality","coarea formula","isocapacity inequality"],"falsifier":"Take $\\alpha=3/2$ and a bounded BV function supported in a neighbourhood of the origin, such as $\\mathbf{1}_{B(0,1)}$, and check whether a sequence of smooth compactly supported functions can approximate it with $\\int|\\nabla_{H_\\alpha}u_h|\\,dx$ converging to $\\|\\nabla_{H_\\alpha}u\\|$; the paper's proof of Theorem 1.5 requires the term $J_2$ to be controlled by $\\varepsilon\\operatorname{Lip}(\\psi,\\Omega)$, and $\\operatorname{Lip}(\\psi,\\Omega)=+\\infty$ for this $\\alpha$. If no such smooth approximation exists, the coarea formula and the Sobolev inequality are not established in the range $1\\le\\alpha<2$.","tokens_in":29172,"feed_emoji":"⚛️","tokens_out":20044,"duration_ms":163374,"temperature":0.7,"pith_summary":"This paper builds a bounded-variation calculus for the $\\alpha$-Hermite operator $H_\\alpha = \\Delta - (\\alpha-1)|x|^\\alpha$, the quantum Hamiltonian of a hydrogen atom fixed at the origin in $\\mathbb{R}^d$. Taking the classical case $\\alpha=1$ as a guide, the authors define the $\\alpha$-Hermite variation, perimeter, and capacity through a generalized gradient whose coefficients carry the weight $x|x|^{(\\alpha-2)/2}$. Their central result is the Sobolev inequality $\\|f\\|_{L^{d/(d-1)}(\\mathbb{R}^d)} \\lesssim \\|\\nabla_{H_\\alpha}f\\|(\\mathbb{R}^d)$ for every $f$ in the $\\alpha$-Hermite BV space, which they prove is equivalent to an isoperimetric inequality and, via the coarea formula, to an isocapacity inequality for sets. Because the induced perimeter fails the symmetry $P_{H_\\alpha}(E)=P_{H_\\alpha}(E^c)$ when $\\alpha>1$, the paper introduces a restricted perimeter that restores this symmetry and uses it to show that sets of finite restricted $\\alpha$-Hermite perimeter have mean curvature in $L^1(\\mathbb{R}^d)$.","feed_headline":"Hydrogen-atom BV theory yields Sobolev and isoperimetric inequalities","feed_subtitle":"The new perimeter and capacity obey the Sobolev, isoperimetric, and isocapacity inequalities of the classical theory.","key_machinery":"The load-bearing object is the generalized gradient $\\nabla_{H_\\alpha}$ assembled from the operators $A^{\\pm}_{i,\\alpha}=\\partial_{x_i}\\pm\\sqrt{\\alpha-1}\\,x_i|x|^{(\\alpha-2)/2}$, which deform Euclidean derivatives by the radial weight $\\psi(x)=x|x|^{(\\alpha-2)/2}$. The argument rides on the coarea formula of Theorem 1.10, $\\|\\nabla_{H_\\alpha}f\\|(\\Omega)\\approx \\int_{-\\infty}^{\\infty}P_{H_\\alpha}(\\{f>t\\},\\Omega)\\,dt$, which turns variation into perimeters of superlevel sets; together with the classical Sobolev inequality and the comparison $|\\nabla f|\\le |\\nabla_{H_\\alpha}f|\\le \\sqrt{2}\\,(|\\nabla f|+\\sqrt{\\alpha-1}\\,|x|^{\\alpha/2}|f|)$, this formula drives the Sobolev inequality and its isoperimetric and isocapacity consequences. The restricted perimeter $\\widetilde{P}_{H_\\alpha}$, defined by imposing one linear constraint on the admissible test functions, repairs the failure $P_{H_\\alpha}(E)\\ne P_{H_\\alpha}(E^c)$ and is the tool used for the mean-curvature result.","core_discovery":"The central claim is Theorem 1.11: for every $f\\in BV_{H_\\alpha}(\\mathbb{R}^d)$ one has $\\|f\\|_{L^{d/(d-1)}(\\mathbb{R}^d)} \\lesssim \\|\\nabla_{H_\\alpha}f\\|(\\mathbb{R}^d)$, and this Sobolev inequality is equivalent to the isoperimetric inequality $|E|^{1-1/d}\\lesssim P_{H_\\alpha}(E)$ for bounded sets of finite $\\alpha$-Hermite perimeter. Theorem 2.10 then derives the isocapacity inequality $|M|^{(d-1)/d}\\lesssim \\operatorname{cap}(M,BV_{H_\\alpha}(\\mathbb{R}^d))$ for compact sets and the converse estimate $\\operatorname{cap}(M,BV_{H_\\alpha}(\\mathbb{R}^d))\\lesssim |M|+P_{H_\\alpha}(M)$ for connected compact sets with smooth boundary, so the $\\alpha$-HBV capacity behaves like the classical BV capacity. The paper also shows this capacity is an outer measure and enjoys the regularity properties spelled out in Theorem 2.3, derives a duality formula expressing it as the supremum of Radon measures in the dual space, and establishes a trace/restriction theorem for measures. Finally, it proves that every set of finite restricted $\\alpha$-Hermite perimeter admits a mean curvature in $L^1(\\mathbb{R}^d)$, extending the classical mean-curvature result for sets of finite perimeter to the asymmetric Hermite setting.","pith_inferences":["The restricted perimeter's constraint (1.16) acts as a calibration condition on the admissible test fields; one could investigate whether it selects the natural surface measure for the weighted geometry, just as calibrated perimeters repair complement asymmetry in other weighted settings.","The equivalence in Theorem 2.10 suggests that the $\\alpha$-HBV isocapacity inequality could be sharpened to an identity splitting volume and perimeter, in the spirit of earlier splitting theorems for sharp Sobolev inequalities; looking for the sharp constant is a direct next step.","A natural extension is to build the $p$-capacity theory for $1<p<\\infty$ using the $\\alpha$-Hermite Sobolev spaces and test whether the trace and isocapacity theorems of Section 2 hold uniformly in $p$.","Since the weight $|x|^{\\alpha/2}$ vanishes at the origin, the boundary case $\\alpha\\to 1^+$ interpolates between Euclidean and increasingly weighted geometries; one could test whether the isoperimetric constant degenerates as $\\alpha$ grows."],"forward_implications":["The $\\alpha$-HBV capacity is an outer measure with the regularity properties of Theorem 2.3, so the standard potential-theoretic machinery applies to it.","The isocapacity inequality $|M|^{(d-1)/d}\\lesssim \\operatorname{cap}(M,BV_{H_\\alpha}(\\mathbb{R}^d))$ means that a compact set of small capacity must have small volume, exactly as in the classical BV theory.","The trace/restriction theorem gives a necessary and sufficient condition on a Radon measure to support the endpoint Sobolev inequality from $BV_{H_\\alpha}$.","Taking $\\alpha=1$ recovers the classical BV theory, with the $\\alpha$-perimeter equal to twice the usual perimeter.","Every set of finite restricted $\\alpha$-Hermite perimeter minimises the variational functional $F_{u,H_\\alpha}$ of (3.1) for some $u\\in L^1(\\mathbb{R}^d)$."],"supporting_citations":[{"why":"supplies the measure-theoretic toolkit: approximation of BV functions, the coarea formula, and the Sobolev inequality used in Theorem 1.11.","marker":"[11]"},{"why":"provides the BV-capacity model and the coarea-type estimate that underpin Theorem 1.10.","marker":"[17]"},{"why":"the classical mean-curvature theorem that Section 3 generalises to the restricted α-Hermite perimeter.","marker":"[5]"},{"why":"supplies the method for proving equivalence between analytic and geometric capacity inequalities in Theorem 2.10.","marker":"[39]"},{"why":"the Gaussian BV-capacity framework whose trace and isocapacity results are adapted to the α-Hermite setting.","marker":"[41]"},{"why":"the α-Hermite Sobolev p-capacity that Proposition 2.6 compares with the new BV capacity.","marker":"[19]"},{"why":"contributes the approximation and coarea argument used in the proof of Theorem 1.10.","marker":"[30]"},{"why":"the classical reference for BV-capacity and weak derivatives that anchors the definitions in Section 2.","marker":"[42]"}],"fun_headline_variants":["α-Hermite BV recovers Sobolev and isoperimetric inequalities","Hydrogen-operator Sobolev inequality implies isoperimetric and isocapacity","Capacity from α-Hermite BV matches classical Sobolev and isoperimetric","α-Hermite sets of finite perimeter admit L1 mean curvature","Sobolev, isoperimetric, isocapacity for α-Hermite BV space"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The approximation theorem that drives the coarea formula and the Sobolev inequality assumes the weight function $\\psi(x)=x\\,|x|^{(\\alpha-2)/2}$ is Lipschitz continuous on the domain, but this fails near the origin for every $1\\le\\alpha<2$.","fun_headline_variants_meta":{"raw":{"variants":["α-Hermite BV recovers Sobolev and isoperimetric inequalities","Hydrogen-operator Sobolev inequality implies isoperimetric and isocapacity","Capacity from α-Hermite BV matches classical Sobolev and isoperimetric","α-Hermite sets of finite perimeter admit L1 mean curvature","Sobolev, isoperimetric, isocapacity for α-Hermite BV space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1578,"prompt_tokens":927,"completion_tokens":651,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":546}},"tokens_in":543,"tokens_out":651,"duration_ms":6152,"temperature":1.0,"reasoning_tokens":546,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:57:04.323532+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\alpha=3/2$ and a bounded BV function supported in a neighbourhood of the origin, such as $\\mathbf{1}_{B(0,1)}$, and check whether a sequence of smooth compactly supported functions can approximate it with $\\int|\\nabla_{H_\\alpha}u_h|\\,dx$ converging to $\\|\\nabla_{H_\\alpha}u\\|$; the paper's proof of Theorem 1.5 requires the term $J_2$ to be controlled by $\\varepsilon\\operatorname{Lip}(\\psi,\\Omega)$, and $\\operatorname{Lip}(\\psi,\\Omega)=+\\infty$ for this $\\alpha$. If no such smooth approximation exists, the coarea formula and the Sobolev inequality are not established in the range $1\\le\\alpha<2$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the measure-theoretic toolkit: approximation of BV functions, the coarea formula, and the Sobolev inequality used in Theorem 1.11."},{"cited_title":"Hakkarainen and J","cited_arxiv_id":null,"evidence_quote":"provides the BV-capacity model and the coarea-type estimate that underpin Theorem 1.10."},{"cited_title":"Gonzalez and I","cited_arxiv_id":null,"evidence_quote":"the classical mean-curvature theorem that Section 3 generalises to the restricted α-Hermite perimeter."},{"cited_title":"Xiao, The sharp Sobolev and isoperimetric inequalit ies split twice, Adv","cited_arxiv_id":null,"evidence_quote":"supplies the method for proving equivalence between analytic and geometric capacity inequalities in Theorem 2.10."},{"cited_title":"Xiao, Gaussian BV Capacity, Adv","cited_arxiv_id":null,"evidence_quote":"the Gaussian BV-capacity framework whose trace and isocapacity results are adapted to the α-Hermite setting."},{"cited_title":"Huang, P","cited_arxiv_id":null,"evidence_quote":"the α-Hermite Sobolev p-capacity that Proposition 2.6 compares with the new BV capacity."},{"cited_title":"Miranda, Jr., Functions of bounded variation on “goo d” metric spaces, J","cited_arxiv_id":null,"evidence_quote":"contributes the approximation and coarea argument used in the proof of Theorem 1.10."},{"cited_title":"Ziemer, Weakly Differentiable Functions, GTM 120, Springer-V erlag, 1989","cited_arxiv_id":null,"evidence_quote":"the classical reference for BV-capacity and weak derivatives that anchors the definitions in Section 2."}],"review_version":1}