The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

Jill Pipher; Linhan Li; Steve Hofmann; Svitlana Mayboroda

REVIEW 2 major objections 3 minor 22 references

The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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 $p\in(1,\infty)$…

desk verdict 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. read the letter →

arxiv 1908.08587 v2 pith:YEJHTR5B submitted 2019-08-22 math.AP

Steve Hofmann , Linhan Li , Svitlana Mayboroda , Jill Pipher This is my paper · ORCID

classification math.AP MSC 35J1535J2542B37

keywords ellipticmeasureDirichletproblemBMOantisymmetriccoefficientsA-infinityweightsCarlesonestimatesquarefunctionestimatesnon-tangentialmaximalHodgedecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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

This 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[Section 3.5, Proposition 3.9] 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 5, smooth approximation argument] 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.

minor comments (3)

[Section 4.2, Lemma 4.2] 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 5, limiting argument] 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.
[General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the A_infinity result is derived from an independently proved Carleson measure estimate; self-citations to [13] are parameter-free inputs, not restatements of the target.

full rationale

The derivation of Theorem 1.1 is self-contained conditional on standard prior results and the companion square-function theory in [13]. The central new content is the Carleson measure estimate Theorem 2.2, proved via Lemma 2.4 in Section 5; no parameter is fitted to the boundary data or to the elliptic measure, and the conclusion (A_infinity / Lp solvability) is never assumed in the proof. The main external inputs, Propositions 3.8-3.10 (square function estimates), Propositions 3.11-3.12 (nontangential maximal estimates), and the Hodge decomposition facts, are separate, parameter-free theorems whose hypotheses are ellipticity and BMO semi-norm bounds, not the target A_infinity property; the fact that several come from the same authors' companion paper [13] does not make them restatements of Theorem 1.1. Lemma 2.1 converts the Carleson estimate into A_infinity by an implication proved in [15]/[18], and the reverse-Holder-to-solvability argument is standard; neither step defines the target in terms of itself. The uniqueness/Fatou argument in Section 6 uses earlier maximal-function estimates but is not needed for existence. Accordingly no circular step can be exhibited; the reliance on Prop. 3.9 is a verification or correctness risk, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants, no invented quantities, and no empirical calibration appear. The theorem is a derivation from standard analysis and cited companion results; the heaviest unproved input is the companion square-function theory [13].

assumptions (5)

standard math Caccioppoli, Poincare-Sobolev, John-Nirenberg, Moser iteration, and reverse Holder inequalities for elliptic and parabolic equations.
Used throughout Sections 3.1-3.4 and in the estimates for the cutoff and the Carleson measure proof.
domain assumption Existence, Harnack, maximum principle, and boundary behavior of elliptic measure for operators with BMO antisymmetric part, from [18].
Section 2 constructs elliptic measure and Lemma 2.1 relies on [18] to reduce A∞ to a Carleson measure bound.
domain assumption Lp square function and Gaussian kernel estimates for the semigroups e^{-t^2 L_||} and e^{-t^2 L*_||}, from companion [13].
Propositions 3.8-3.10 and [13] Theorems 4.8-4.9 are imported without proof and used repeatedly in the proof of Lemma 2.4 in Section 5.
domain assumption Solution of the Kato square root problem for elliptic operators with unbounded BMO antisymmetric coefficients, from [7].
Cited as a prior result supporting the square function estimates used in the proof overview and Section 3.5.
domain assumption Reverse Holder and kernel estimates for elliptic measure from [12] and [17] that pass from A∞ to Lp solvability.
Used in Section 2, equations (2.4)-(2.5), to conclude existence of solutions from the A∞ property.

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Cite this review

Pith. "Pith review of The Dirichlet problem for elliptic operators having a BMO anti-symmetric part." pith.science (2026). https://pith.science/paper/YEJHTR5B

@misc{pith2026190808587,
  author       = {Pith},
  title        = {Pith review of: The Dirichlet problem for elliptic operators having a BMO anti-symmetric part},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YEJHTR5B}},
  note         = {Machine review of arXiv:1908.08587}
}

abstract

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation ${\rm div}(A\nabla u)=0$ in the upper half-space $(x,t)\in\mathbb{R}^{n+1}_+$ is uniquely solvable when $n\ge2$ and the boundary data is in $L^p(\mathbb{R}^n,dx)$ for some $p\in (1,\infty)$. This result is equivalent to saying that the elliptic measure associated to $L$ belongs to the $A_\infty$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [13]

$L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

Steve Hofmann, Linhan Li, Svitlana Mayboroda, and Jill Pipher. Lp theory for the square roots and square functions of elliptic operators having a BM O anti-symmetric part. arXiv preprint arXiv:1908.01030, 2019

work page Pith review arXiv 1908

[2] [1]

The Diri chlet problem for second order parabolic operators in divergence form

Pascal Auscher, Moritz Egert, and Kaj Nystr¨ om. The Diri chlet problem for second order parabolic operators in divergence form. Journal de l’Ecole Polytechnique , 5:407–441, 2018

work page 2018

[3] [2]

The solution of the Kato square root problem for second order elliptic operators on Rn

Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIn tosh, and Philippe Tchamitchian. The solution of the Kato square root problem for second order elliptic operators on Rn. Annals of mathematics , 156(2):633–654, 2002

work page 2002

[4] [3]

Extrapolation of Carleson measures and the analyticity of Kato’s square-r oot operators

Pascal Auscher, Steve Hofmann, John L Lewis, and Philipp e Tchamitchian. Extrapolation of Carleson measures and the analyticity of Kato’s square-r oot operators. Acta mathematica, 187(2):161–190, 2001

work page 2001

[5] [4]

Caﬀarelli, E

L. Caﬀarelli, E. Fabes, S. Mortola, and S. Salsa. Boundar y behavior of nonnegative solutions of elliptic operators in divergence form. Indiana University Mathematics Journal , 30(4):621– 640, 1981

work page 1981

[6] [5]

Comp letely singular elliptic-harmonic measures

Luis A Caﬀarelli, Eugene B Fabes, and Carlos E Kenig. Comp letely singular elliptic-harmonic measures. Indiana University Mathematics Journal , 30(6):917–924, 1981

work page 1981

[7] [6]

Coifman, P.L

R. Coifman, P.L. Lions, Y. Meyer, and S. Semmes. Compensa ted compactness and Hardy spaces. J. Math. Pures Appl. , 72(9):247–286, 1993. 60 STEVE HOFMANN, LINHAN LI, SVITLANA MAYBORODA, AND JILL PI PHER

work page 1993

[8] [7]

Kato square root prob lem with unbounded leading coeﬃcients

Luis Escauriaza and Steve Hofmann. Kato square root prob lem with unbounded leading coeﬃcients. Proceedings of the American Mathematical Society , 146(12):5295–5310, 2018

work page 2018