{"id":"69fb292d-e0f8-4f21-8138-6aeb517be92e","arxiv_id":"1908.08587","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For t-independent divergence form operators with an L∞ elliptic symmetric part and a BMO antisymmetric part, the elliptic measure is A∞ and the Lp Dirichlet problem is uniquely solvable in the upper half-space for n≥2.","lead":"This paper proves that elliptic equations with coefficients whose symmetric part is bounded and whose antisymmetric part has bounded mean oscillation still admit unique boundary solutions with L^p data in the upper half-space. It matters because this is the first boundary-regularity result for such unbounded coefficients, a case where many classical arguments fail.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Most load-bearing risk is the unproved endpoint square-function estimate in Prop. 3.9: it is the only source of the p>2 control of BMO terms, and Section 5 uses it at p=2+ε0 in the estimates for I12 and II^a_1122.","rationale":"The reader's weakest_assumption correctly identifies the imported square-function estimates, especially Propositions 3.8-3.10 from [13], as the most fragile part of the proof. I have sharpened this to Proposition 3.9, because unlike Propositions 3.8 and 3.10 it has a restricted range 1<p≤2+ε0 and is used at the endpoint p=2+ε0 in Section 5 precisely where BMO coefficients require higher integrability. The rest of the paper, including the smooth approximation and uniqueness arguments, contains some abbreviated but standard-looking steps; no independent mathematical inconsistency was found in the main Carleson-measure chain once the imported square-function estimates are granted. The reader's CONDITIONAL verdict already reflects this dependency, so my stress-test does not move the verdict. The proposed check, an independent verification of [13, Prop. 6.3] and its two applications in Section 5, would settle whether the concern actually lands.","tokens_in":50082,"tokens_out":16530,"duration_ms":169683,"concrete_test":"Independently re-derive [13, Prop. 6.3] for p=2+ε0 with constants tracked as functions of λ0, Λ0, and n. Then redo the two estimates in Section 5 labelled I12 and II^a_1122 using only the re-derived statement, checking in particular that the exponent p=2α/(2−α)=2+ε0 satisfies the hypotheses and that the implicit constants are uniform in the cube Q. If the re-derivation requires an extra hypothesis (e.g., small BMO norm, bounded coefficients, or a weaker range p<2+ε0), the concern lands; if it reproduces the statement with uniform constants, the proof step is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.5 states Propositions 3.8-3.10 with proofs deferred to [13]. Of these, Proposition 3.9 (the p≤2+ε0 square-function estimate) is the one whose restricted range matters: in Section 5, the estimates labelled I12 and II^a_1122 explicitly set α/(2−α)=(2+ε0)/2 and invoke Proposition 3.9 with p=2+ε0 to control products of BMO coefficients with ∇u and semigroup terms. This is exactly the higher integrability that replaces bounded coefficients after the John-Nirenberg inequality. The manuscript does not reproduce the argument, and [13] is a companion arXiv preprint by the same authors; no independent verification is provided here. If the constant in [13, Prop. 6.3] depends on the BMO norm in a way that is not uniform over cubes, or if the range p≤2+ε0 is not actually available for the operator L_|| with unbounded antisymmetric part, then the estimates for I12 and II^a_1122 fail and Lemma 2.4 is not established. Since Theorem 1.1 is obtained as a corollary of Lemma 2.4 via Lemmas 2.1 and 2.3, this is a load-bearing external input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that, for a real, t-independent (n+1) x (n+1) coefficient matrix A whose symmetric part is uniformly elliptic and essentially bounded and whose antisymmetric part is in BMO(R^n), the L^p Dirichlet problem for L = -div(A∇) in the upper half-space is uniquely solvable for some p in (1,∞), with n ≥ 2. The authors show that this is equivalent to the elliptic measure belonging to A_∞ with respect to Lebesgue measure. The proof is organized as a reduction to a Carleson measure estimate: Theorem 2.2 is the main quantitative estimate, Lemma 2.3 allows restriction to a large subset F of each boundary cube, and Lemma 2.4 is the core estimate proved in Section 5 using a Hodge decomposition, an adapted cutoff function, and L^p estimates for square functions and non-tangential maximal functions associated with the n-dimensional operators L_|| and L*_||. The paper also proves a Fatou-type theorem (Theorem 1.2) that yields uniqueness of the Dirichlet problem and is stronger than mere uniqueness. Several technical inputs, especially the square-function estimates in Propositions 3.8-3.10, are taken from the authors' companion paper [13], with proofs deferred there.","tokens_in":50354,"tokens_out":8361,"duration_ms":85279,"significance":"If the main theorem is correct, this is a substantial advance beyond the bounded-coefficient theory: it is the first absolute-continuity result for elliptic measure when the antisymmetric part of the coefficient matrix is allowed to be unbounded, in the sharp BMO class. The paper is carefully written and the main Carleson measure argument is detailed, with all constants tracked through the ellipticity constant and the BMO seminorm; no fitted constants or post-hoc exclusions appear. The Fatou-type uniqueness theorem is also a useful independent contribution. The principal weakness is that the proof leans on several imported L^p square-function estimates from the companion preprint [13], and the most delicate of these, Proposition 3.9, is used at the endpoint range p = 2+ε0 in a load-bearing way in Section 5. The manuscript's central claim is therefore conditional on the correctness of those external results, and the paper would be strengthened by including their proofs or by a clear verification of the required range and uniformity.","major_comments":[{"comment":"Proposition 3.9 (the L^p square-function estimate for t∇∂_t e^{-t^2L}F, 1 < p ≤ 2+ε0) is stated with proof deferred to the companion paper [13]. This estimate is load-bearing for Theorem 1.1: in Section 5, the estimate for I12 explicitly sets α/(2−α) = (2+ε0)/2 and applies Proposition 3.9 at p = 2+ε0, and the later estimate for III113 is described as being bounded 'by the same method of estimating I12.' Since [13] is an arXiv preprint by the same authors and its proof is not reproduced or summarized here, the main theorem rests on an unverified external input. I request that the authors either include a complete proof of Proposition 3.9 (or a detailed sketch of the argument), or provide a precise citation to a published version with a statement of the range p ≤ 2+ε0 and of uniformity of the constant with respect to the BMO seminorm. Without this, the Carleson measure estimate leading to Theorem 1.1 is not established within the present manuscript.","section":"Section 3.5, Proposition 3.9"},{"comment":"The reduction to smooth coefficients uses a mollification A_δ = ξ_δ * A_0 with ξ_δ(X) = δ^{-n-1} ξ(X/δ), where X is written as (x,t) in R^{n+1}. Since A_0 is t-independent, full convolution in R^{n+1} would generally make A_δ depend on t, which would conflict with the t-independence used in the reduction to L_0 and in the semigroup arguments. The authors should clarify that the convolution is taken only in the x-variable, or otherwise explain why the resulting operator remains t-independent. This point is technical but directly relevant to the validity of the limiting argument that completes the proof of Lemma 2.4.","section":"Section 5, smooth approximation argument"}],"minor_comments":[{"comment":"In the proof of Lemma 4.2, the text refers to 'Propostion 3.11, Propostion 3.12, and Proposition 4.1'; the last item should be Lemma 4.1, and 'Propostion' should be 'Proposition'.","section":"Section 4.2, Lemma 4.2"},{"comment":"In the smooth approximation argument, the authors invoke a reverse Hölder inequality for ∇u without giving a reference or a proof. Since this inequality is used to justify convergence of the Dirichlet energies in the limit δ → 0, a citation to the relevant result (e.g., from [18]) would improve clarity.","section":"Section 5, limiting argument"},{"comment":"The paper would benefit from a short list of notation, since several symbols (for example, the integrated non-tangential maximal function ~N^α and the dyadic grid D^η_k in Section 5) are introduced locally and used over many pages.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the heavy reliance on the companion preprint [13], especially Proposition 3.9. If the editor can verify that [13] has been accepted for publication or that its proofs are already independently available, the present manuscript would be considerably strengthened. Otherwise, I recommend asking the authors to provide the proof of Proposition 3.9 or a detailed outline with explicit verification of the p-range and BMO-uniformity. The notation issue in the smooth approximation (full convolution vs. x-convolution) should also be resolved before final acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the first A∞/Lp Dirichlet result for t-independent elliptic operators with unbounded BMO antisymmetric part. The main theorem is genuinely new: it removes a boundedness barrier that had persisted through the bounded-coefficient theory. The paper does the job honestly. The proof is a reduction to a Carleson measure estimate (Lemma 2.4), and the new tools—a W^{1,2+ε} Hodge decomposition, semigroup square-function estimates, and a Fatou-type uniqueness theorem (Theorem 1.2)—are real. I read carefully for fitted constants or post-hoc exclusions, and I don't see any.\n\nThe main caveat is the one the stress-test flags: Proposition 3.9, the p≤2+ε0 square-function estimate, is imported from the companion paper [13]. It is genuinely load-bearing. In Section 5, the estimate for I12 explicitly sets α/(2−α)=(2+ε0)/2 and applies Proposition 3.9 at p=2+ε0; the argument for III_111 does the same. This is exactly the higher integrability that controls the BMO coefficients once boundedness is lost. If that proposition has a gap, Lemma 2.4 fails and so does Theorem 1.1. The reliance is out in the open, and [13] is a separate paper by the same authors, not reproduced here. That is a legitimate dependency, not a flaw in itself, but any referee needs to verify [13] before signing off. The stress-test note slightly misreads one spot: II^a_1122 cites Proposition 3.8, not 3.9, and Proposition 3.8 holds for all 1<p<∞. The vulnerable call is I12 (and its echo in III_111), so the concern is real but narrow.\n\nThis paper is for harmonic-analysis-minded PDE readers. It deserves a serious referee: the argument is long, the dependency on the companion paper is heavy, but the result is important and the presentation is transparent. I would send it to peer review rather than desk reject.","headline":"First A∞/Lp Dirichlet result for elliptic operators with unbounded BMO antisymmetric part—a genuine advance, but the proof leans on an endpoint square-function estimate deferred to a companion paper that deserves a careful look.","tokens_in":50887,"tokens_out":3827,"would_cite":true,"duration_ms":34932,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J15","35J25","42B37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for real $t$-independent coefficient matrices with bounded elliptic symmetric part and antisymmetric part in BMO, the $L^p$ Dirichlet problem in the upper half-space is uniquely solvable for some $p\\in(1,\\infty)$…","keywords":["elliptic measure","Dirichlet problem","BMO antisymmetric coefficients","A-infinity weights","Carleson measure estimate","square function estimates","non-tangential maximal function","Hodge decomposition"],"falsifier":"Construct a real t-independent matrix A satisfying (1.1)\\textendash(1.2) and a bounded weak solution u with $\\|u\\|_{L^\\infty}\\le1$ for which $\\sup_Q |Q|^{-1}\\int_0^{l(Q)}\\int_Q |\\nabla u(x,t)|^2\\,t\\,dx\\,dt$ is infinite; that would make Theorem 2.2 false and thereby Theorem 1.1. An explicit or numerical example in which the elliptic measure is singular to Lebesgue measure would also settle the question negatively.","tokens_in":49891,"feed_emoji":"📐","tokens_out":9099,"duration_ms":80972,"temperature":0.7,"pith_summary":"This paper establishes the first boundary well-posedness result for divergence-form elliptic operators with coefficients that are not bounded. The central claim is that for a real matrix $A=A(x)$ whose symmetric part is uniformly elliptic and bounded, and whose antisymmetric part belongs to the John\\textendash Nirenberg space $\\mathrm{BMO}$ (bounded mean oscillation), the $L^p$ Dirichlet problem for $L=-\\operatorname{div}(A\\nabla u)=0$ in the upper half-space $\\mathbb{R}^{n+1}_+$ is uniquely solvable for some $p\\in(1,\\infty)$ when $n\\ge2$. By a known equivalence, this says the elliptic measure associated to $L$ lies in the $A_\\infty$ class with respect to Lebesgue measure, a quantitative form of absolute continuity. The paper also proves a Fatou-type theorem showing that any weak solution with $L^p$ non-tangential maximal function has a.e. non-tangential limits and is a Poisson integral, which implies uniqueness.","feed_headline":"Unbounded BMO drift yields absolutely continuous elliptic measure","feed_subtitle":"First proof that Lp boundary data produce unique solutions for some p>1 even with unbounded coefficients.","key_machinery":"The central mechanism is the Carleson measure estimate for bounded weak solutions, combined with a carefully chosen \\\"good set\\\" $F\\subset Q$ of density at least $999/1000$. On $F$, several auxiliary maximal functions built from the ellipticized heat semigroups $e^{-t^2L_\\parallel}$, $e^{-t^2L_\\parallel^*}$ and from two Hodge-decomposition potentials $\\phi,\\tilde\\phi$ are bounded by a fixed constant $\\kappa_0$. A sawtooth domain over $F$ and a cutoff function $\\Psi$ localize all integrals; the proof repeatedly uses the antisymmetry of the BMO part, the John\\textendash Nirenberg inequality, and imported $L^p$ square-function and non-tangential maximal function estimates. Lemma 2.4 packages the main estimate, and the bootstrap $J\\le(\\sigma+c\\eta)J+\\tilde c|Q|$ with small $\\sigma,\\eta$ yields the uniform Carleson bound.","core_discovery":"On its own terms, the paper's discovery is that the Carleson measure estimate $$\\sup_{Q\\subset\\mathbb{R}^n}\\frac1{|Q|}\\$int_0^{{l(Q)}}$\\int_Q |\\nabla u(x,t)|^2\\,t\\,dx\\,dt\\le C$$ holds for every bounded weak solution $u$ of $L$ with $\\|u\\|_{L^\\infty}\\le1$, with $C$ depending only on dimension, ellipticity, and the BMO norm. The proof shows that this estimate implies $\\omega^{X_Q}\\in A_\\infty(Q)$ for every cube, and $A_\\infty$ weights give the reverse H\\\"older inequality that yields solvability of $(D)_p$ for $p\\ge q'$. The argument works with a modified operator $L_0$ whose BMO coefficients are recentered by cube averages, uses a new $W^{1,2+\\epsilon}$ Hodge decomposition and semigroup estimates, and reduces the Carleson estimate to a bootstrap inequality $J_{\\eta,\\epsilon}\\le(\\sigma+c\\eta)J_{\\eta,\\epsilon}+\\tilde c|Q|$, which gives the desired bound after letting $\\epsilon\\to0$. Uniqueness is obtained by representing any solution with $Nu\\in L^p$ as an elliptic-measure Poisson integral, using the non-tangential maximal function bound for the Poisson kernel.","pith_inferences":["The paper leaves open, but the proof's scale-invariant constants suggest, that the same $A_\\infty$ conclusion should extend to Lipschitz graph domains via the standard change of variables; the paper only states the flat half-space case.","The paper does not attempt to identify the range of admissible $p$ in dimension $2$; the available two-dimensional tools make it plausible that one can pin down the exact range in terms of the ellipticity and BMO constants.","A natural testable extension is the parabolic analogue: equations $\\partial_t u+\\operatorname{div}(A\\nabla u)=0$ with antisymmetric part in $L^\\infty(\\mathrm{BMO})$, which the introduction names as the parabolic counterpart of this elliptic result.","The method's reliance on $L^{2+\\epsilon}$ Hodge decomposition suggests the BMO condition could be relaxed to suitably localized oscillation classes while preserving the constants, though the paper does not address this."],"forward_implications":["For every real $t$-independent matrix satisfying (1.1)\\textendash(1.2), the $L^p$ Dirichlet problem $(D)_p$ is well posed for some $p\\in(1,\\infty)$, with non-tangential convergence and an $L^p$ non-tangential maximal function bound.","The elliptic measure $\\omega^X$ is quantitatively mutually absolutely continuous with Lebesgue measure on $\\mathbb{R}^n$; in particular its density satisfies a reverse H\\\"older inequality.","Theorem 1.2 gives a Poisson representation for arbitrary weak solutions with $N u\\in L^p$: the non-tangential limit exists a.e. and belongs to $L^p$, so uniqueness holds. This theorem does not require $t$-independence.","The constants depend only on dimension, ellipticity, and the BMO seminorm, so scale-invariant results follow: the same $p$ and $A_\\infty$ parameters work for all cubes.","The known reduction from $A_\\infty$ to solvability means the result includes a range of exponents $p$ (all $p\\ge q'$), not merely the single exponent used in the proof."],"supporting_citations":[{"why":"supplies the Lp square-function and square-root estimates for operators with a BMO antisymmetric part, used in Propositions 3.8\\textendash 3.10 and in the definition of F.","marker":"[13]"},{"why":"solves the Kato square root problem for unbounded leading coefficients, providing the L2 estimate for the square root of L that underlies the semigroup estimates.","marker":"[7]"},{"why":"constructs elliptic measure and proves boundary H\\\"older regularity for operators with BMO antisymmetric part, giving the framework that turns Carleson bounds into A-infinity.","marker":"[18]"},{"why":"provides the square function/non-tangential maximal function strategy for non-symmetric bounded t-independent operators that the present proof adapts.","marker":"[10]"},{"why":"is the source of the criterion that a Carleson measure estimate on the gradient yields the A-infinity property of elliptic measure.","marker":"[15]"},{"why":"introduces the divergence-free drift framework and the weak-solution interpretation that makes BMO antisymmetric coefficients admissible.","marker":"[22]"},{"why":"shows how reverse H\\\"older estimates for the elliptic kernel imply Lp Dirichlet solvability, used in the existence part.","marker":"[12]"}],"fun_headline_variants":["BMO anti-symmetric part forces elliptic measure into A_infinity","Unbounded coefficients, yet elliptic measure is absolutely continuous","First L^p solvability for elliptic operators with BMO drift","BMO drift: elliptic measure A_infinity despite unboundedness","Elliptic measure absolutely continuous with unbounded BMO coefficients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument leans on previously established technical estimates for the same class of operators, in particular Lp square-function and square-root bounds; if any of those companion results fails, the Carleson measure bound and hence the main theorem do not follow.","fun_headline_variants_meta":{"raw":{"variants":["BMO anti-symmetric part forces elliptic measure into A_infinity","Unbounded coefficients, yet elliptic measure is absolutely continuous","First L^p solvability for elliptic operators with BMO drift","BMO drift: elliptic measure A_infinity despite unboundedness","Elliptic measure absolutely continuous with unbounded BMO coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000285,"raw_usage":{"total_tokens":1694,"prompt_tokens":978,"completion_tokens":716,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":626}},"tokens_in":594,"tokens_out":716,"duration_ms":6606,"temperature":1.0,"reasoning_tokens":626,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:35:34.066150+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a real t-independent matrix A satisfying (1.1)\\textendash(1.2) and a bounded weak solution u with $\\|u\\|_{L^\\infty}\\le1$ for which $\\sup_Q |Q|^{-1}\\int_0^{l(Q)}\\int_Q |\\nabla u(x,t)|^2\\,t\\,dx\\,dt$ is infinite; that would make Theorem 2.2 false and thereby Theorem 1.1. An explicit or numerical example in which the elliptic measure is singular to Lebesgue measure would also settle the question negatively.","supporting_citations":[{"cited_title":"$L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part","cited_arxiv_id":"1908.01030","evidence_quote":"supplies the Lp square-function and square-root estimates for operators with a BMO antisymmetric part, used in Propositions 3.8\\textendash 3.10 and in the definition of F."},{"cited_title":"Kato square root prob lem with unbounded leading coefficients","cited_arxiv_id":null,"evidence_quote":"solves the Kato square root problem for unbounded leading coefficients, providing the L2 estimate for the square root of L that underlies the semigroup estimates."},{"cited_title":"Boundary behavior of solutio ns of elliptic operators in divergence form with a BMO anti-symmetric part","cited_arxiv_id":null,"evidence_quote":"constructs elliptic measure and proves boundary H\\\"older regularity for operators with BMO antisymmetric part, giving the framework that turns Carleson bounds into A-infinity."},{"cited_title":"Square function/non- tangential maximal function estimates and the Dirichlet pr oblem for non-symmetric elliptic operators","cited_arxiv_id":null,"evidence_quote":"provides the square function/non-tangential maximal function strategy for non-symmetric bounded t-independent operators that the present proof adapts."},{"cited_title":"Square f unctions and the A∞ property of elliptic measures","cited_arxiv_id":null,"evidence_quote":"is the source of the criterion that a Carleson measure estimate on the gradient yields the A-infinity property of elliptic measure."},{"cited_title":"On divergence-free drifts","cited_arxiv_id":null,"evidence_quote":"introduces the divergence-free drift framework and the weak-solution interpretation that makes BMO antisymmetric coefficients admissible."},{"cited_title":"Carleson mea sure estimates and the Dirichlet problem for degenerate elliptic equations","cited_arxiv_id":null,"evidence_quote":"shows how reverse H\\\"older estimates for the elliptic kernel imply Lp Dirichlet solvability, used in the existence part."}],"review_version":1}