{"id":"a066d755-290c-4a4f-92dc-294520139122","arxiv_id":"2506.07752","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The logarithmic equilibrium measure on C^{1,α} curves is absolutely continuous with respect to length measure, a new result for d≥3.","lead":"This paper proves that the logarithmic equilibrium measure on a compact set in R^d is absolutely continuous with respect to arclength on any part that is locally a C^{1,α} curve. For dimensions d≥3 this is the first general structural result, and it resolves even the question of whether the measure's support can be lower-dimensional.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The local minimum principle (Cor. 2.4) is the linchpin of Thm 1.3, but its proof via Appendix B borrows a weak Lebesgue point identity from [45] that is not verified for the logarithmic kernel.","rationale":"The reader correctly identifies Corollary 2.4 as the weakest assumption. My stress-test sharpens the concern: the problem is not the localization of the Ahlfors regularity condition, which Appendix B handles plausibly, but the fact that the quoted result [45, Proposition 2.7] is a statement about Riesz s-potentials, whereas the manuscript needs it for the logarithmic potential. The paper gives no derivation of the weak Lebesgue point identity for the log kernel, and the classical theory for superharmonic kernels does not obviously cover d≥3, where −log|x| is not superharmonic away from its singularity. This is a genuine soft spot because every step of §2.3 after the reduction—the continuity of Uµ on sptµ∩B(3r), the membership of µ1 in M(Γ), and the equality U^Γµ = L on the support—depends on Corollary 2.4. The rest of the argument (Sections 3–7) is long but coherent, and I found no comparable gap in the bootstrapping or operator estimates. Since the concern is about a missing verification of a key cited input rather than a demonstrated falsehood, the appropriate verdict is CONDITIONAL: the theorem should be accepted only after the log-kernel version of the minimum principle is either proved directly or shown to follow from [45]. If such a proof is supplied, the paper's central claim would stand with the current argument.","tokens_in":66449,"tokens_out":32879,"duration_ms":419091,"concrete_test":"Read the statements of [45, Theorem 2.5 and Proposition 2.7] and determine whether they cover the logarithmic kernel −log|x−y|, not just Riesz kernels |x−y|^{-s}. If they are Riesz-only, attempt a direct proof of the weak Lebesgue point identity (B.2) for the log kernel on a compact Ahlfors 1-regular set, e.g. by splitting the logarithmic kernel into an L^1 piece and a bounded piece and using Ahlfors regularity; if no proof can be given, or if a measure on a C^{1,α} curve is found for which (B.2) fails, then Corollary 2.4 and hence the proof of Theorem 1.3 are unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The deduction of Theorem 1.3 from Theorem 1.9 in §2.3 needs U µ ≥ E_log(µ) everywhere on γ∩B(3r), not just approximately everywhere. Corollary 2.4 is used for exactly this upgrade, and its proof in Appendix B reduces to the weak s-Lebesgue point identity (B.2) for Uµ, attributed to [45, Proposition 2.7]. The paper does not check that [45] applies to the logarithmic kernel. [45] is a minimum principle for Riesz s-potentials |x−y|^{-s} with s∈(0,d); Theorem 2.3 of the present paper states the same result with U the logarithmic potential and s∈(0,d], but the logarithmic kernel is not a Riesz s-kernel and no limiting argument is supplied. Since Theorem 2.2 only gives U^µ ≥ E approximately everywhere, the proof's key reduction would collapse if (B.2) failed for the log kernel on an Ahlfors regular curve. Appendix B asserts (B.2) for 'Uµ' without verification; the gap is not a matter of localizing the Ahlfors regularity, but of the kernel class covered by [45].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the logarithmic equilibrium measure µ on a compact set γ⊂R^d. The main result, Theorem 1.3, states that µ is absolutely continuous with respect to arclength on the set γ_reg of points where γ is locally a C^{1,α} graph, α>0. The proof reduces Theorem 1.3 to a one-dimensional statement, Theorem 1.9, asserting that on a small-Lipschitz C^{1,α} graph Γ, any measure µ∈M(R) whose graph potential U^Γµ coincides with a Lipschitz function on sptµ∩I0 and is ≥ that function on I0 is absolutely continuous on compact subintervals of I0, with L^p density. The proof of Theorem 1.9 splits U^Γµ=Pµ+Rµ into a convex principal part and an α-Hölder remainder, then uses the equality/inequality hypotheses plus Proposition 3.14 to obtain Hölder regularity of U^Γµ. A bootstrap in §4.1 converts, via L^p estimates for fractional Laplacians and Proposition 3.1, the available Frostman exponents of µ into higher ones, eventually reaching exponent >1−α, which yields L^p integrability of µ. The remaining sections prove the required L^p invertibility of the operators T_β=∆^{(1−β)/2}U^Γ∆^{β/2} by Calderón–Zygmund theory and a comparison with the flat graph. The paper also contains a critical Remark 1.1 questioning part of the proof of [35, Theorem 2.7], though it does not rely on that theorem.","tokens_in":66675,"tokens_out":19321,"duration_ms":239288,"significance":"If the stated theorem is correct, it is a substantial advance: for d≥3 it is the first result showing that the logarithmic equilibrium measure on C^∞ curves is absolutely continuous with respect to arclength, and it answers the previously open question of whether the support has positive dimension in that setting. The technical apparatus is impressive and largely self-contained: the decomposition U=P+R with explicit convexity and Hölder estimates, the bootstrapping scheme with matching Frostman exponents, and the uniform L^p estimates for truncated operators are coherent and detailed. I did not find independent errors in Sections 3–7 once the minimum principle is granted. The one load-bearing gap concerns the local minimum principle underlying the reduction of Theorem 1.3 to Theorem 1.9, detailed in the major comments; this is localized and appears repairable.","major_comments":[{"comment":"The proof of Corollary 2.4 is not valid as written for the logarithmic potential. Theorem B.1 invokes [45, Proposition 2.7] for the weak s-Lebesgue point identity (B.2), but [45] concerns Riesz s-potentials |x−y|^{-s}, whereas the application in §2.3 requires the same conclusion for Uµ(x)=∫−log|Γ(x)−Γ(y)|dµ(y). The logarithmic kernel is not an s-Riesz kernel, no limiting argument is supplied, and the proof in Appendix B does not verify (B.2) for this kernel. Since Theorem 2.2 only supplies the lower bound Uµ≥E_log(µ) approximately everywhere, this gap is load-bearing: without an everywhere lower bound on γ∩B(3r), the equality Uµ=E_log(µ) on sptµ∩B(3r) used to verify the hypotheses of Theorem 1.9 is not established. The gap appears fixable by a direct proof of the weak Lebesgue point property for -log|x−y| on Ahlfors 1-regular sets, but it must be supplied.","section":"Appendix B / Corollary 2.4"},{"comment":"The statement of Theorem 2.3 is ambiguous about which kernel U denotes. In §2.1, U is the logarithmic potential, but the cited result [45, Theorem 2.5] is a minimum principle for Riesz s-potentials, and §1.3 uses [45] in exactly that Riesz capacity. If Theorem 2.3 is intended for the logarithmic potential, it is not a special case of [45]; if it is intended for the Riesz s-potential, then Corollary 2.4 is not applicable to the logarithmic equilibrium measure in §2.3. Either way, the manuscript needs a consistent statement and proof of the minimum principle for the kernel actually used.","section":"Theorem 2.3 / Corollary 2.4"}],"minor_comments":[{"comment":"The stated range γ∈(0,1−α] is empty when α=1; the case α=1 should be handled separately or the notation should be adjusted.","section":"Lemma 3.13"},{"comment":"The auxiliary function log_+ is defined as max{log,1}, which conflicts with the standard usage max{log,0}; the text should use a different symbol or explicitly explain the deliberate cutoff.","section":"Proposition 7.2"},{"comment":"The critique of [35, Theorem 2.7] is not used later in the paper; the authors should state explicitly that no later argument depends on that theorem, so the remark is not read as an unsupported assertion affecting the proof.","section":"Remark 1.1"},{"comment":"Several displayed formulas contain typesetting/OCR artifacts (for example, the notation for averaged balls and the truncated logarithm functions) that should be cleaned up in the final version.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the referee report focuses on Corollary 2.4; this is the same concern identified in the stress-test note, and I believe it lands. If the authors supply a direct proof of the weak Lebesgue point property for the logarithmic kernel on Ahlfors 1-regular sets, or an appropriate citation, the main theorem should be acceptable. I do not see a need for further external review beyond the usual."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The take on this paper: it's a serious and mostly convincing proof of a genuinely new result—for d≥3, the logarithmic equilibrium measure on a C^{1,α} curve is absolutely continuous with respect to arclength, and previously even positive dimension of the support was unknown. The d=2 graph case is acknowledged as classical via harmonic measure. The bootstrapping scheme (Section 4.1) and the operator estimates (Sections 5–7) look coherent, and the paper is unusually honest about what it does and doesn't show, e.g., Remark 1.1 on a point in [35] that the authors did not understand.\n\nWhere I'd push back is not the main architecture but one specific load-bearing step: the proof of Theorem 1.3 from Theorem 1.9 in Section 2.3 needs the minimum principle upgrade from 'approximately everywhere' to 'everywhere' (Cor 2.4). Appendix B derives Cor 2.4 from [45, Proposition 2.7], which the authors claim gives the weak s-Lebesgue point identity (B.2) for the logarithmic potential Uμ. But [45] is about Riesz s-potentials |x−y|^{−s}; the logarithmic kernel is not a Riesz kernel, and no limiting argument is supplied. This is not a cosmetic gap: without an everywhere lower bound on spt μ∩B(3r), you only get Uμ = E_log quasi-everywhere on the support, which is not enough for equality everywhere in the hypotheses of Theorem 1.9. I think the gap is repairable—the log kernel is locally integrable and the required Lebesgue point argument should carry over—but the paper needs to either cite a log-kernel version or give the few lines of proof.\n\nMinor issues: a handful of typos, and the remark about [35] is candid but unresolved. The citation pattern looks fair; the only self-citation is [42], used for an elementary lemma, which is fine.\n\nWho this is for: people working in equilibrium measure, potential theory, and quantitative rectifiability. It deserves a serious referee; the potential significance is high. My recommendation: send it to review, but flag Cor 2.4/Appendix B as the part the referee should check carefully. Ask for a repaired justification of (B.2) for the log kernel before acceptance.\n\nBest,","headline":"A substantial new theorem in logarithmic potential theory, but the passage from the Riesz-kernel minimum principle to the logarithmic kernel is not fully justified and needs referee scrutiny.","tokens_in":67227,"tokens_out":8186,"would_cite":true,"duration_ms":93423,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["31B15","31A15","42B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"On the $C^{1,\\alpha}$-regular part of a compact set, the logarithmic equilibrium measure is absolutely continuous with respect to arclength — in $d \\geq 3$ this was open even for $C^{\\infty}$ graphs.","keywords":["logarithmic equilibrium measure","logarithmic potential","C^{1,\\alpha} curves","absolute continuity","length measure","fractional Laplacian","Frostman estimates","Calderón-Zygmund operators"],"falsifier":"Pick a compact smooth arc in $\\mathbb{R}^3$, for instance the parabola graph $\\gamma = \\{(t, t^2, 0) : |t| \\leq 1\\}$, and compute the logarithmic equilibrium measure on a compact subinterval of the interior, numerically or by rigorous discretisation. The theorem predicts a density that lies in every $L^p$ space and dimension bounds $\\mu(B(x,r)) \\lesssim r^{\\kappa}$ for every $\\kappa < 1$; a subinterval where the dimension exponent drops strictly below 1, or where any positive mass sits on an arclength-null set, would refute Theorem 1.3. A cheaper probe targets the weakest premise directly: build a compact set that is $C^{1,\\alpha}$ at one point but not Ahlfors regular in any neighbourhood, and test whether the approximate lower bound $U\\mu \\geq E_{\\log}(\\mu)$ holds at every regular point — Corollary 2.4 asserts it always does, and the reduction to Theorem 1.9 collapses if that fails.","tokens_in":66234,"feed_emoji":"📏","tokens_out":16302,"duration_ms":149222,"temperature":0.7,"pith_summary":"The paper proves Theorem 1.3: for a compact set $\\gamma \\subset \\mathbb{R}^d$ of positive logarithmic capacity, the logarithmic equilibrium measure — the unique probability measure on $\\gamma$ minimizing the energy $\\int\\!\\int -\\log|x-y|\\,d\\mu(x)\\,d\\mu(y)$ — is absolutely continuous with respect to arclength on the part of $\\gamma$ that is locally a $C^{1,\\alpha}$ graph, $\\alpha > 0$. In dimensions $d \\geq 3$ this is the first general structure theorem for this measure, and it resolves the standing question of whether the support of an equilibrium measure on a smooth curve can have positive Hausdorff dimension: on a compact $C^{1,\\alpha}$ curve the support has full dimension one. In the plane the statement is a classical consequence of identifying the equilibrium measure with harmonic measure of the complement, so the proof is built for the genuinely higher-dimensional case. The technical core, Theorem 1.9, shows that on a small-slope graph a measure whose graph potential equals a Lipschitz function on its support and dominates it everywhere must be given by a density that is locally in every $L^p$ space.","feed_headline":"Equilibrium measure on smooth curves is absolutely continuous","feed_subtitle":"The log-capacitary charge on a C^{1,α} curve is a density against arclength — new even for C^∞ graphs in d ≥ 3.","key_machinery":"The proof runs on three objects. First, the graph potential is decomposed as $U^{\\Gamma}\\mu = P\\mu + R\\mu$, where the principal part $P\\mu$ (built from a smoothed gradient of $A$) is convex outside the support of $\\mu$, and the remainder $R\\mu$ is $\\alpha$-Hölder; $R\\mu$ becomes $(\\alpha+\\kappa)$-Hölder once $\\mu$ satisfies the dimension estimate $\\mu(B(x,r)) \\lesssim r^{\\kappa}$. Convexity plus the equality/inequality hypotheses (Proposition 3.14) force $U^{\\Gamma}\\mu$ itself to be Hölder continuous. Second, a family of fractional-Laplacian operators $T_\\beta^{\\Gamma} = \\Delta^{(1-\\beta)/2} U^{\\Gamma} \\Delta^{\\beta/2}$, $\\beta \\in [0,1]$, are shown by Calderón–Zygmund theory — kernels of the form $k(\\Gamma(x) - \\Gamma(y))$ multiplied by difference quotients $(A_i(x) - A_i(y))/(x - y)$ — to be uniformly bounded on $L^p$ and, for small $\\mathrm{Lip}\\,A$, invertible (Theorem 7.1); this converts Hölder regularity of the potential into dimension (Frostman-type) estimates on $\\mu$. Third, a bootstrap iterates the dimension exponent through the improved remainder estimates until it passes $1 - \\alpha$, at which point $U^{\\Gamma}\\mu$ is Lipschitz and the $\\beta = 0$ case of the same operator theory yields $\\mu$ locally in $L^p$. The initial ignition is the local minimum principle of Corollary 2.4, which upgrades the classical approximate lower bound $U\\mu \\geq E_{\\log}(\\mu)$ to an everywhere bound on locally Ahlfors-regular pieces of the curve, so that the potential is actually constant on the relevant supports.","core_discovery":"The central claim is Theorem 1.3: if $\\gamma \\subset \\mathbb{R}^d$ is a compact set with positive logarithmic capacity and $\\mu$ is its logarithmic equilibrium measure, then $\\mu$ restricted to $\\gamma_{\\mathrm{reg}}$ — the set of points where $\\gamma$ is locally the graph of a $C^{1,\\alpha}$ function $\\mathbb{R} \\to \\mathbb{R}^{d-1}$ for some $\\alpha > 0$ — is absolutely continuous with respect to the length measure. Equivalently, on any compact $C^{1,\\alpha}$ curve, or a finite union of such curves, the equilibrium measure has a density with respect to arclength, and the support of the measure has Hausdorff dimension one on the regular part. The theorem is deduced from the local statement Theorem 1.9, which the paper proves in full: for a graph $\\Gamma(x) = (x, A(x))$ with $A \\in C^{1,\\alpha}$ and sufficiently small Lipschitz constant, if a measure $\\mu$ on $\\mathbb{R}$ has the property that its logarithmic graph potential $U^{\\Gamma}\\mu$ agrees with a Lipschitz function $L$ on the support of $\\mu$ inside an interval $I_0$ and is bounded below by $L$ throughout $I_0$, then $\\mu$ is absolutely continuous on compact subintervals of $I_0$, with density in $L^p$ for every finite $p$. For $d = 2$ this local statement can be recovered from classical harmonic measure theory for graphs, but for $d \\geq 3$ both Theorem 1.9 and Theorem 1.3 are new, including for $C^{\\infty}$ graphs.","pith_inferences":["The kernel split 'convex principal part plus Hölder remainder' does not use anything special about the logarithmic kernel beyond its singularity, so a similar bootstrap plausibly works for Riesz kernels of order $s < d-2$ on $C^{1,\\alpha}$ curves — a regime where no structural results currently exist and which the paper explicitly leaves open.","The local density of the equilibrium measure is predicted on any smooth arc in $\\mathbb{R}^3$ to obey the dimension bounds $\\mu(B(x,r)) \\lesssim r^{\\kappa}$ for every $\\kappa < 1$; that quantitative fingerprint could be checked by a numerical computation on a generic smooth arc, e.g. a parabolic graph, without computing the density itself.","The small-slope hypothesis $\\delta = \\delta(p, \\alpha, d)$ enters only through the $L^p$ invertibility of $T_\\beta^{\\Gamma}$ (Theorem 7.1); if that invertibility survives without the smallness assumption, the same proof would extend absolute continuity to $C^{1,\\alpha}$ curves of arbitrary slope, and possibly to Lipschitz curves.","In the plane the theorem is a shadow of harmonic-measure theory, as the authors note; the higher-dimensional mechanism suggests that the right structural theory for equilibrium measures on low-dimensional sets in $\\mathbb{R}^d$ is built from local potential decompositions and operator invertibility rather than from subharmonicity of the kernel."],"forward_implications":["On a compact $C^{1,\\alpha}$ curve, or a finite union of such curves, the logarithmic equilibrium measure has a density against arclength; in $d \\geq 3$ this answers the previously open question of whether the support has positive Hausdorff dimension — it has full dimension one on the regular part.","Theorem 1.9's local $L^p$ conclusion means the density may be unbounded — as on the interval, where $d\\mu = dx/(\\pi\\sqrt{1-x^2})$ — but it can never concentrate a singular component on any sub-arc.","The asymptotic distribution of logarithmically optimal $N$-point configurations (Fekete points) on $C^{1,\\alpha}$ curves is therefore governed by an absolutely continuous equilibrium measure.","Absolute continuity is not mutual, even when the whole curve is regular: a unit circle with any smooth curve attached inside it carries its equilibrium measure entirely on the circle (Remark 1.4).","The small-slope graph machinery is the announced template (Section 1.4) for treating $(m-1)$-Riesz equilibrium measures on $m$-dimensional $C^{1,\\alpha}$ surfaces by the same bootstrap."],"supporting_citations":[{"why":"Supplies the minimum principle for potentials (its Theorem 2.5) from which the local upgrade of approximate to everywhere lower bounds, Corollary 2.4, is derived in Appendix B.","marker":"[45]"},{"why":"Provides existence of equilibrium measures, the variational lower/upper bounds for their potentials (Theorem 2.2), and the continuity principle (Theorem 2.5) used in the reduction to graphs.","marker":"[39]"},{"why":"Grounds the classical potential theory of logarithmic and Riesz kernels, including the subharmonicity facts that explain why the new argument is only needed for $d \\geq 3$.","marker":"[40]"},{"why":"Supplies the pointwise formulae for fractional Laplacians, homogeneous distributions, Mihlin multipliers, and the three lines lemma that underpin the operators $T_\\beta^{\\Gamma}$.","marker":"[33]"},{"why":"Identifies $\\Delta^{-\\beta/2}$ with convolution against the Riesz kernel, connecting $L^p$ bounds of fractional Laplacians to dimension estimates on $\\mu$ in Proposition 2.8.","marker":"[54]"},{"why":"Gives the $L^2$ boundedness of truncated singular integrals with odd kernels on Lipschitz graphs, the engine of the uniform $L^p$ estimates for the truncated operators.","marker":"[49]"},{"why":"Provides the stable-kernel version of the $T1$ machinery used to extend boundedness to kernels carrying Lipschitz difference-quotient factors in Proposition 6.7.","marker":"[20]"}],"fun_headline_variants":["Log-capacity measure on curves is a density","Smooth curves: equilibrium measure now known to be a density","Equilibrium measure on C^{1,α} curves has arclength density","In high dimensions, equilibrium measure on curves is a density"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the local minimum principle of Corollary 2.4: a lower bound on the logarithmic potential that holds 'almost everywhere' on a regular piece of the curve (outside a capacity-zero set) must hold at every point of that piece; if this upgrade fails, the potential need not equal the Lipschitz function on the whole support, and the bootstrap never starts.","fun_headline_variants_meta":{"raw":{"variants":["Log-capacity measure on curves is a density","Smooth curves: equilibrium measure now known to be a density","Equilibrium measure on C^{1,α} curves has arclength density","In high dimensions, equilibrium measure on curves is a density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000863,"raw_usage":{"total_tokens":3813,"prompt_tokens":1086,"completion_tokens":2727,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":702,"completion_tokens_details":{"reasoning_tokens":2656}},"tokens_in":702,"tokens_out":2727,"duration_ms":21241,"temperature":1.0,"reasoning_tokens":2656,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:28:30.506067+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a compact smooth arc in $\\mathbb{R}^3$, for instance the parabola graph $\\gamma = \\{(t, t^2, 0) : |t| \\leq 1\\}$, and compute the logarithmic equilibrium measure on a compact subinterval of the interior, numerically or by rigorous discretisation. The theorem predicts a density that lies in every $L^p$ space and dimension bounds $\\mu(B(x,r)) \\lesssim r^{\\kappa}$ for every $\\kappa < 1$; a subinterval where the dimension exponent drops strictly below 1, or where any positive mass sits on an arclength-null set, would refute Theorem 1.3. A cheaper probe targets the weakest premise directly: build a compact set that is $C^{1,\\alpha}$ at one point but not Ahlfors regular in any neighbourhood, and test whether the approximate lower bound $U\\mu \\geq E_{\\log}(\\mu)$ holds at every regular point — Corollary 2.4 asserts it always does, and the reduction to Theorem 1.9 collapses if that fails.","supporting_citations":[{"cited_title":"Reznikov, E","cited_arxiv_id":null,"evidence_quote":"Supplies the minimum principle for potentials (its Theorem 2.5) from which the local upgrade of approximate to everywhere lower bounds, Corollary 2.4, is derived in Appendix B."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides existence of equilibrium measures, the variational lower/upper bounds for their potentials (Theorem 2.2), and the continuity principle (Theorem 2.5) used in the reduction to graphs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Grounds the classical potential theory of logarithmic and Riesz kernels, including the subharmonicity facts that explain why the new argument is only needed for $d \\geq 3$."},{"cited_title":"Wolff.Lectures on harmonic analysis, volume 29 ofUniversity Lecture Series","cited_arxiv_id":null,"evidence_quote":"Identifies $\\Delta^{-\\beta/2}$ with convolution against the Riesz kernel, connecting $L^p$ bounds of fractional Laplacians to dimension estimates on $\\mu$ in Proposition 2.8."},{"cited_title":"Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogo- nality.Proc","cited_arxiv_id":null,"evidence_quote":"Gives the $L^2$ boundedness of truncated singular integrals with odd kernels on Lipschitz graphs, the engine of the uniform $L^p$ estimates for the truncated operators."},{"cited_title":"Springer, Berlin, Heidelberg, 1991","cited_arxiv_id":null,"evidence_quote":"Provides the stable-kernel version of the $T1$ machinery used to extend boundedness to kernels carrying Lipschitz difference-quotient factors in Proposition 6.7."}],"review_version":1}