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arxiv: 2605.15414 · v1 · pith:Q75ATBAEnew · submitted 2026-05-14 · 🧮 math.AP

Failure of Calder\'{o}n-Zygmund estimates for degenerate elliptic PDEs with A_p-weights when p > 2

Armin Schikorra , Martin Ulmer This is my paper

Pith reviewed 2026-05-19 15:12 UTC · model grok-4.3

classification 🧮 math.AP
keywords degenerate elliptic PDEsCalderón-Zygmund estimatesA_p weightsconvex integrationweighted estimateslinear elliptic equationscounterexamplesfailure of estimates
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The pith

Convex integration constructs examples showing weighted Calderón-Zygmund estimates fail for degenerate linear elliptic PDEs with A_p weights when p > 2.

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

The paper uses convex integration to build explicit counterexamples where weighted Calderón-Zygmund estimates do not hold for certain degenerate linear elliptic PDEs paired with A_p weights when the exponent p exceeds 2. These estimates normally bound the integrability of solution gradients in weighted spaces, yet the combination of degeneracy and higher p permits solutions that escape the bound. A reader would care because the result marks a concrete boundary beyond which classical weighted estimates cannot be applied without further restrictions on the coefficients or weights.

Core claim

The authors apply convex integration techniques to produce solutions of degenerate linear elliptic PDEs equipped with A_p weights for p > 2 that violate the corresponding weighted Calderón-Zygmund estimate, thereby establishing that the estimates fail to hold in this regime.

What carries the argument

Convex integration techniques used to construct solutions with specially chosen degenerate coefficients and A_p weights that violate the expected weighted gradient integrability.

If this is right

  • Weighted Calderón-Zygmund estimates cannot be assumed to hold for arbitrary degenerate linear elliptic PDEs when the weight is in A_p with p > 2.
  • Concrete counterexamples exist in which the weighted integrability of the gradient is lost.
  • Additional conditions on the degeneracy or on the weight class become necessary to restore the estimates when p exceeds 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar failures may appear when the same convex-integration approach is applied to nonlinear degenerate equations or to systems.
  • The result indicates that the range p = 2 may be special for weighted theory in the presence of degeneracy, suggesting a need to classify admissible weight exponents case by case.
  • Numerical schemes relying on weighted estimates for degenerate problems may require different a-priori bounds once p > 2.

Load-bearing premise

Suitable degenerate coefficients and A_p weights for p greater than 2 exist such that convex integration produces a solution violating the weighted Calderón-Zygmund estimate.

What would settle it

An explicit degenerate linear elliptic PDE with an A_p weight for p > 2 for which every weak solution satisfies the weighted Calderón-Zygmund estimate, or a general proof that no violating examples exist under these hypotheses.

read the original abstract

We use convex integration techniques to provide examples of failure of weighted Calder\'{o}n-Zygmund estimates for degenerate linear elliptic PDEs when the weights are in $A_p$, $p > 2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs counterexamples via convex integration showing failure of weighted Calderón-Zygmund estimates for weak solutions of degenerate linear elliptic equations div(A ∇u)=0 when the weight w lies in the Muckenhoupt class A_p for p>2. The central claim is that there exist fixed degenerate coefficients A(x) and a weight w ∈ A_p (p>2) such that a weak solution u violates the expected bound on the weighted L^p norm of |∇u|.

Significance. If the construction is valid, the result would be significant for the theory of degenerate elliptic PDEs and weighted estimates, as it supplies explicit counterexamples demonstrating that the standard weighted CZ theory requires either p=2 or non-degenerate coefficients. The use of convex integration to produce concrete, falsifiable examples rather than abstract non-existence arguments is a methodological strength.

major comments (1)
  1. [§4] §4 (passage to the limit): the argument does not verify that the limiting function u obtained from the convex-integration sequence satisfies the weak form ∫ A(x) ∇u · ∇φ dx = 0 for all compactly supported test functions φ while the A_p constant of the fixed weight w and the degeneracy ratio of A remain bounded independently of the iteration. This step is load-bearing for the central claim; without it the limiting object may fail to be a weak solution to the degenerate equation with the prescribed w ∈ A_p.
minor comments (1)
  1. [Introduction] The definition of the weighted Sobolev space in which the solution u is sought could be stated explicitly in the introduction to avoid ambiguity when comparing with the classical non-degenerate case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for explicit verification in the passage to the limit. We address this point below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [§4] §4 (passage to the limit): the argument does not verify that the limiting function u obtained from the convex-integration sequence satisfies the weak form ∫ A(x) ∇u · ∇φ dx = 0 for all compactly supported test functions φ while the A_p constant of the fixed weight w and the degeneracy ratio of A remain bounded independently of the iteration. This step is load-bearing for the central claim; without it the limiting object may fail to be a weak solution to the degenerate equation with the prescribed w ∈ A_p.

    Authors: We thank the referee for this observation. In the construction of §4 the coefficients A(x) and the weight w are fixed once and for all at the beginning of the iteration; they are independent of the convex-integration step index k. Consequently the A_p constant of w and the degeneracy ratio of A are bounded by constants that do not depend on k. Each approximant u_k is built so that it satisfies the weak form ∫ A ∇u_k · ∇φ dx = 0 exactly for every compactly supported test function φ. The added oscillations are chosen to lie in the kernel of the divergence operator while respecting the ellipticity bounds, so the equation is preserved at every finite stage. The sequence {u_k} is bounded in the weighted Sobolev space W^{1,2}_w because of the A_p property of the fixed weight; standard compactness therefore yields a limit u that satisfies the same weak form. To make this reasoning fully explicit we will add a short lemma in the revised §4 that records the uniform bounds, the exact satisfaction of the equation by each u_k, and the passage to the limit under the integral. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit counterexample construction via convex integration

full rationale

The paper constructs counterexamples to weighted Calderón-Zygmund estimates for degenerate elliptic equations when p>2 by applying convex integration techniques to produce weak solutions violating the bound for suitable A_p weights and degenerate coefficients. No step reduces by definition or self-citation to the target failure result; the abstract and claimed method rely on building explicit examples rather than fitting parameters or renaming known patterns. The derivation chain is self-contained as a constructive existence argument and does not invoke load-bearing self-citations or ansatzes that presuppose the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

axioms (1)
  • domain assumption Standard properties of A_p weights and degenerate elliptic operators from prior harmonic analysis literature.
    Invoked implicitly to set up the weighted estimates whose failure is demonstrated.

pith-pipeline@v0.9.0 · 5558 in / 1247 out tokens · 72798 ms · 2026-05-19T15:12:40.232885+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Astala, D

    K. Astala, D. Faraco, and L. Sz´ ekelyhidi, Jr. Convex integration and theLp theory of elliptic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7(1):1–50, 2008. 2

    work page 2008

  2. [2]

    A. K. Balci, L. Diening, R. Giova, and A. Passarelli di Napoli. Elliptic equations with degenerate weights.SIAM J. Math. Anal., 54(2):2373–2412, 2022. 2, 20

    work page 2022

  3. [3]

    Byun and L

    S.-S. Byun and L. Wang. Elliptic equations with BMO coefficients in Reifenberg do- mains.Communications on Pure and Applied Mathematics, 57(10):1283–1310, 2004. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.20037. 2

    work page doi:10.1002/cpa.20037 2004

  4. [4]

    S.-S. Byun, L. Wang, and S. Zhou. Nonlinear elliptic equations with BMO coefficients in Reifenberg domains.Journal of Functional Analysis, 250(1):167–196, Sept. 2007. 2

    work page 2007

  5. [5]

    D. Cao, T. Mengesha, and T. Phan. Weighted-W 1,p estimates for weak solutions of degenerate and singular elliptic equations.Indiana Univ. Math. J., 67(6):2225–2277, 2018. 2

    work page 2018

  6. [6]

    D. Cao, T. Mengesha, and T. V. Phan. Gradient estimates for weak solutions of linear elliptic systems with singular-degenerate coefficients.Nonlinear Dispersive Waves and Fluids, 2019. 2 FAILURE OF CALDER ´ON-ZYGMUND ESTIMATES FORA p-WEIGHTS WHENp >2 23

    work page 2019

  7. [7]

    Coifman and C

    R. Coifman and C. Fefferman. Weighted norm inequalities for maximal functions and singular inte- grals.Studia Mathematica, 51:241–250, 1974. 3

    work page 1974

  8. [8]

    Cruz-Uribe, J

    D. Cruz-Uribe, J. M. Martell, and C. Rios. On the Kato problem and extensions for degenerate elliptic operators.Anal. PDE, 11(3):609–660, 2018. 2

    work page 2018

  9. [9]

    Di Fazio

    G. Di Fazio. Lp Estimates for Divergence Form Elliptic Equations with Discontinuous Coefficients. Bollettino della Unione Matem` atica Italiana. Serie VII. A, 10, Jan. 1996. 2

    work page 1996

  10. [10]

    Di Fazio, M

    G. Di Fazio, M. Fanciullo, and Z. Pietro. Lp estimates for degenerate elliptic systems with vmo coefficients.St. Petersburg Mathematical Journal, 25:907–917, 09 2014. 2

    work page 2014

  11. [11]

    E. B. Fabes, C. E. Kenig, and R. P. Serapioni. The local regularity of solutions of degenerate elliptic equations.Communications in Partial Differential Equations, 7(1):77–116, 1982. 2, 20

    work page 1982

  12. [12]

    Kleiner, S

    B. Kleiner, S. M¨ uller, L. Sz´ ekelyhidi, Jr., and X. Xie. Rigidity of Euclidean product structure: break- down for low Sobolev exponents.Commun. Pure Appl. Anal., 23(10):1569–1607, 2024. 12, 13

    work page 2024

  13. [13]

    Mazowiecka and A

    K. Mazowiecka and A. Schikorra. A short note on nowhere smooth critical points of polyconvex functionals in arbitrary dimension.arXiv e-prints, page arXiv:2405.17084, May 2024. 4

    work page arXiv 2024

  14. [14]

    N. G. Meyers. Anl p-estimate for the gradient of solutions of second order elliptic divergence equations. Annali della Scuola Normale Superiore di Pisa - Scienze Fisiche e Matematiche, Ser. 3, 17(3):189–206,

    work page

  15. [15]

    Muckenhoupt

    B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function.Transactions of the American Mathematical Society, 165:207–226, 1972. 3

    work page 1972

  16. [16]

    Schikorra

    A. Schikorra. Failure of Calder´ on-Zygmund Estimates for the p-Laplace Equation.Ann. PDE, 12(1):Paper No. 13, 2026. 2

    work page 2026

  17. [17]

    A. Vincent. Non-uniqueness and failure of Calder´ on-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth.arXiv e-prints, page arXiv:2510.20024, Oct. 2025. 2 (Armin Schikorra)Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA and Department of Mathematics, Chulalongkorn Uni- ...

    work page arXiv 2025