pith. sign in

arxiv: 2410.23801 · v4 · submitted 2024-10-31 · 🧮 math.AP · math.CA

The L^p regularity problem for parabolic operators

Martin Dindo\v{s} , Linhan Li , Jill Pipher This is my paper

Pith reviewed 2026-05-23 18:58 UTC · model grok-4.3

classification 🧮 math.AP math.CA
keywords paraboliccarlesoncasecoefficientsproblemquestionregularityelliptic
0
0 comments X

The pith

The L^p regularity problem for parabolic operators with Carleson coefficients is solvable for p in (1, p0) for some p0 > 1.

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

This paper shows that the regularity problem for the parabolic equation with divergence form operator is solvable in L^p spaces for p close to 1. The coefficients are only assumed to be elliptic, bounded and measurable but must satisfy a Carleson-type condition that measures their oscillation in a scale invariant manner. This result holds on Lipschitz cylinders in space-time and resolves the question even when the Carleson norm is small.

Core claim

Under the assumptions that the coefficient matrix A is elliptic with bounded measurable coefficients satisfying a parabolic Carleson condition, the regularity problem for -∂_t u + div(A ∇u) = 0 is solvable in the range (1, p0) for some p0 > 1 on Lipschitz cylinders.

What carries the argument

The natural parabolic Carleson condition on the coefficients, a scale-invariant bound on their oscillations over cylinders, which enables the L^p estimates for the solution gradient.

Load-bearing premise

The coefficients satisfy a natural parabolic Carleson condition in addition to being elliptic with bounded measurable entries.

What would settle it

A specific example of an elliptic bounded measurable coefficient matrix satisfying the Carleson condition for which no p>1 allows solvability of the regularity problem would falsify the result.

Figures

Figures reproduced from arXiv: 2410.23801 by Jill Pipher, Linhan Li, Martin Dindo\v{s}.

Figure 1
Figure 1. Figure 1: Novel cones in the (t, xn) plane. To explain this definition we compare (10.6) and (10.8). When the time variable is not present there is no difference between these definitions, and thus there is no reason to consider (10.8). This is due to the fact that in this case the boundary of the cone (10.6) is a union of straight lines of slope a −1 originating from its vertex and intersections of such cones resul… view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of domains in the (x, xn) plane. Our goal is to establish that ∇w ∈ L 2 (Ω0). Given what we know about u and v we clearly have ∇w = ∇u − ∇v = 0 when t ≤ −11. Also since (11.23) and (11.25) holds when t ≥ 22 we obtain by using Lemma 3.10 and the exponential decay (Lemma 11.1) that ∥∇w∥L2(Ω0∩{t≥22}) ≤ ∥∇u∥L2(Ω0∩{t≥22}) + ∥∇v∥L2(Ω0∩{t≥22}) ≲ ∥f∥L˙ p 1,1/2 (∂Ω). On the component of the domain Ω0 {(x ′ ,… view at source ↗
read the original abstract

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small. In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.

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

0 major / 2 minor

Summary. The manuscript claims to fully resolve the L^p regularity problem for the parabolic PDE −∂_t u + div(A ∇ u) = 0 on a Lipschitz cylinder O × R. Under the assumptions that A is elliptic with bounded measurable coefficients satisfying a parabolic Carleson condition (the natural parabolic analog of the DKP condition), the authors prove that the regularity problem is solvable for all p ∈ (1, p_0) for some p_0 > 1. The result is asserted to hold even when the Carleson norm is small, a case previously open; the proof strategy is motivated by one of the recent elliptic resolutions but must overcome new parabolic difficulties.

Significance. If the central proof is correct, the result would constitute a substantial advance in the theory of parabolic equations with rough coefficients. It completes the parabolic counterpart to the recent elliptic resolutions of the regularity problem (one of which is due to two of the present authors together with Hofmann) and supplies the first proof even in the small-perturbation regime. The manuscript explicitly identifies the new parabolic obstacles and claims to surmount them, which, if verified, would be a notable technical contribution.

minor comments (2)
  1. The abstract states that the result holds 'for some p0 > 1' but does not indicate whether an explicit value or dependence on the Carleson norm is obtained; a brief remark in the introduction on the size of p0 would be helpful for context.
  2. Notation for the parabolic Carleson condition is introduced without an equation number in the abstract; assigning a numbered display equation when the condition is first written would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the significance of resolving the parabolic L^p regularity problem, including in the small Carleson norm regime, is recognized as a substantial advance.

Circularity Check

0 steps flagged

Minor self-citation to elliptic precedent; parabolic proof independent

full rationale

The manuscript cites prior elliptic results (one involving two of the present authors) only as motivation and explicitly notes new parabolic challenges that are addressed by a distinct argument. No load-bearing step reduces to a self-citation chain, fitted input, or definitional equivalence. The central claim is a direct proof under ellipticity, bounded measurable coefficients, and the parabolic Carleson condition, with no internal reduction of the stated solvability result to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on standard ellipticity, boundedness, and the Carleson condition as the key hypothesis; no free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption Ellipticity and bounded measurability of coefficients
    Standard assumption for the PDE to be well-posed, invoked in the problem statement.
  • domain assumption Lipschitz cylinder domain
    The setting for the regularity problem.
  • domain assumption Carleson condition on coefficients
    The central hypothesis enabling the result.

pith-pipeline@v0.9.0 · 5744 in / 1164 out tokens · 55197 ms · 2026-05-23T18:58:07.981696+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    Aronson, Non-negative solutions of linear parabolic equations , Ann

    D. Aronson, Non-negative solutions of linear parabolic equations , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. 13, 15

    work page 1968

  2. [2]

    Azzam, Semi-uniform domains and the A∞ property of harmonic measure

    J. Azzam, Semi-uniform domains and the A∞ property of harmonic measure . Int. Math. Res. Not. (2021) no. 9, 6717–6771. 4

    work page 2021

  3. [3]

    Auscher M

    P. Auscher M. Egert, K. Nystr¨ om,L2 well-posedness of boundary value problems for parabolic systems with measurable coefficients, Journal of the EMS, 2020, 22 (9), pp.2943-3058. 5, 9, 10 110 DINDO ˇS, LI, AND PIPHER

    work page 2020

  4. [4]

    Azzam, S

    J. Azzam, S. Hofmann, J. M. Martell, M. Mourgoglou, and X. Tolsa, Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem. Invent. Math. 222 (2020), no. 3, 881–993. 4

    work page 2020

  5. [5]

    Brown, Layer Potentials and Boundary Value Problems for the Heat Equation on Lipschitz Cylinders , Thesis (Ph.D.)–University of Minnesota ProQuest LLC, Ann Arbor, MI, 1987

    R. Brown, Layer Potentials and Boundary Value Problems for the Heat Equation on Lipschitz Cylinders , Thesis (Ph.D.)–University of Minnesota ProQuest LLC, Ann Arbor, MI, 1987. 94 pp. 5, 13

    work page 1987

  6. [6]

    Brown, The method of layer potentials for the heat equation in Lipschitz cylinders , Amer

    R. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders , Amer. J. Math. 111 (1989), no. 2, 339–379. 5, 13

    work page 1989

  7. [7]

    Castro, S

    A. Castro, S. Rodr´ ıguez-L´ opez, W. Staubach,L2-solvability of the Dirichlet, Neumann and regularity problems for parabolic equations with time-independent H¨ older-continuous coefficients , Trans. Amer. Math. Soc. 370 (2018), no. 1, 265–319. 5, 13

    work page 2018

  8. [8]

    Coifman, Y

    R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), no.2, 304–335. 17

    work page 1985

  9. [9]

    Dahlberg, Poisson semigroups and singular integrals , Proc

    B. Dahlberg, Poisson semigroups and singular integrals , Proc. Amer. Math. Soc. 97 (1) (1976) 41–48. 2, 96

    work page 1976

  10. [10]

    Z. Dai, J. Feneuil, S. Mayboroda, The regularity problem in domains with lower dimensional boundaries , J. Funct. Anal. 284 (2023), no. 11, Paper No. 109903, 72 pp. 7, 73

    work page 2023

  11. [11]

    Dindoˇ s,On the regularity problem for parabolic operators and the role of half time derivative , arXiv preprint arXiv:2308.12936 (2023)

    M. Dindoˇ s,On the regularity problem for parabolic operators and the role of half time derivative , arXiv preprint arXiv:2308.12936 (2023). 4, 5, 9, 13, 106

    work page arXiv 2023

  12. [12]

    Dindoˇ s, L

    M. Dindoˇ s, L. Dyer, Parabolic regularity and Dirichlet boundary value problems , Nonlinear Anal. 185 (2019), 247–263. 8, 16

    work page 2019

  13. [13]

    Dindoˇ s, L

    M. Dindoˇ s, L. Dyer, S. Hwang, Parabolic Lp Dirichlet boundary value problem and VMO-type time- varying domains, Anal. PDE 13 (2020), no. 4, 1221–1268. 4, 91

    work page 2020

  14. [14]

    Dindoˇ s, S

    M. Dindoˇ s, S. Hofmann, J. Pipher,Regularity and Neumann problems for operators with real coefficients satisfying Carleson condition, J. Funct. Anal. 285 (2023), no. 6, Paper No. 110024, 32 pp. 2, 4, 5, 6, 17, 24, 73, 77

    work page 2023

  15. [15]

    Dindoˇ s, S

    M. Dindoˇ s, S. Hwang,The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition, Rev. Mat. Iberoam. 34 (2018), no. 2, 767–810. 4, 12, 18, 44, 88, 91

    work page 2018

  16. [16]

    Dindoˇ s, S

    M. Dindoˇ s, S. Hwang, M. Mitrea,The Lp Dirichlet boundary problem for second order elliptic systems with rough coefficients, Trans. Amer. Math. Soc. 374 (2021), no. 5, 3659–3701. 73, 76, 77, 86

    work page 2021

  17. [17]

    Dindoˇ s, J

    M. Dindoˇ s, J. Pipher, Boundary value problems for elliptic operators satisfying Carleson condition, Vietnam J. Math. 52 (2024), no. 3, 627—673. 4

    work page 2024

  18. [18]

    Dindoˇ s, S

    M. Dindoˇ s, S. Petermichl, J. Pipher,The Lp Dirichlet problem for second order elliptic operators and a p-adapted square function, Jour. F. Anal. 249 no.2 (2007), 372–392. 3, 4, 18, 22, 100

    work page 2007

  19. [19]

    Dindoˇ s, S

    M. Dindoˇ s, S. Petermichl, J. Pipher,BMO solvability and the A∞ condition for second order parabolic operators, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 34 (2017), no. 5, 1155–1180. 4, 91

    work page 2017

  20. [20]

    Dindoˇ s, J

    M. Dindoˇ s, J. Pipher, D. Rule,The Boundary value problems for second order elliptic operators satisfying Carleson condition, Comm. Pure Appl. Math. 70 (2017), no. 7, 1316–1365. 4, 6

    work page 2017

  21. [21]

    Dindoˇ s, D

    M. Dindoˇ s, D. Rule, Elliptic equations in a plane satisfying the Carleson measure condition , Revista Mathematica Iberoamericana 26 no.3 (2010), 1013–1034. 4

    work page 2010

  22. [22]

    Dindoˇ s, E

    M. Dindoˇ s, E. S¨ atterqvist,A relation between the Dirichlet and the Regularity problem for Parabolic equations, arXiv:2409.09197. 5, 14, 16, 89, 90, 91, 92, 93

    work page arXiv

  23. [23]

    Fabes, M

    E. Fabes, M. Jodeit, Lp boundary value problems for parabolic equations , Bull. Amer. Math. Soc. 74 (1968), 1098–1102 7

    work page 1968

  24. [24]

    Fabes, N

    E. Fabes, N. Rivi` ere,Singular integrals with mixed homogeneity , Studia Math. 27 (1966), 19–38. 8

    work page 1966

  25. [25]

    Fabes, N

    E. Fabes, N. Rivi` ere, Symbolic calculus of kernels with mixed homogeneity , Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 106–127, AMS, Providence, RI, 1967 8

    work page 1966

  26. [26]

    Fabes, N

    E. Fabes, N. Rivi` ere, Dirichlet and Neumann problems for the heat equation in C 1-cylinders, Proc. Sympos. Pure Math., XXXV, Part 2, AMS Providence, RI, 1979, pp. 179–196. 5

    work page 1979

  27. [27]

    Fefferman, E

    C. Fefferman, E. Stein, H p spaces of several variables , Acta Mat. 129 (1972, 137–193

    work page 1972

  28. [28]

    J. Feneuil, An alternative proof of the Lp-regularity problem for Dahlberg-Kenig-Pipher operators on Rn +, arXiv preprint arXiv:2310.00645 17 THE Lp REGULARITY PROBLEM FOR PARABOLIC OPERATORS 111

    work page arXiv

  29. [29]

    Feneuil, Carleson perturbations of locally Lipschitz elliptic operators, arXiv preprint arXiv:2408.10061 16

    J. Feneuil, Carleson perturbations of locally Lipschitz elliptic operators, arXiv preprint arXiv:2408.10061 16

    work page arXiv

  30. [30]

    Feneuil and L

    J. Feneuil and L. Li, A Green function characterization of uniformly rectifiable sets of any codimension. Adv. Math. 430 (2023), Paper No. 109220, 45 pp. 4

    work page 2023

  31. [31]

    Feneuil, L

    J. Feneuil, L. Li, S. Mayboroda, Green functions and smooth distances , Math. Ann. 389 (2024), no.3, 2637–2727. 18

    work page 2024

  32. [32]

    Gilbarg, N

    D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Classics Math. Springer- Verlag, Berlin, 2001, xiv+517 pp. ISBN: 3-540-41160-7 29

    work page 2001

  33. [33]

    Grafakos, Modern Fourier analysis , Third edition, Grad

    L. Grafakos, Modern Fourier analysis , Third edition, Grad. Texts in Math., 250 Springer, New York,

    work page

  34. [34]

    Hofmann, J

    S. Hofmann, J. Lewis, L2 solvability and representation by caloric layer potentials in time-varying do- mains, Ann. of Math. (2) 144 (1996), no. 2, 349–420. 7

    work page 1996

  35. [35]

    Hofmann, J

    S. Hofmann, J. Lewis, The Lp regularity problem for the heat equation in non-cylindrical domains , Hofmann, Steven; Lewis, John L. Illinois J. Math. 43 (1999), no. 4, 752–769. 7

    work page 1999

  36. [36]

    Hofmann, J

    S. Hofmann, J. Lewis, The Dirichlet problem for parabolic operators with singular drift terms , Mem. Amer. Math. Soc. 151 no. 719 (2001), pp. viii+113. 6, 14, 15, 96

    work page 2001

  37. [37]

    Hofmann, J.M

    S. Hofmann, J.M. Martell, S. Mayboroda, Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains . Anal. PDE (to appear). 4

    work page

  38. [38]

    Hofmann, J.M

    S. Hofmann, J.M. Martell, S. Mayboroda, T. Toro, Z. Zihui, Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Geom. Funct. Anal. 31 (2021), no. 2, 325–401. 4

    work page 2021

  39. [39]

    Hofmann, J.M

    S. Hofmann, J.M. Martell, and T. Toro, A∞ implies NTA for a class of variable coefficient elliptic operators. J. Differential Equations 263 (2017), no.10, 6147–6188. 4

    work page 2017

  40. [40]

    Kenig, H

    C. Kenig, H. Koch, J. Pipher, T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations , Adv. Math. 153 (2000), no. 2, 231–298. 73

    work page 2000

  41. [41]

    Kenig, J

    C. Kenig, J. Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients , Invent. Math. 113 no. 3 (1993), 447–509. 4

    work page 1993

  42. [42]

    Kenig, J

    C. Kenig, J. Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients. II, A celebration of John F. Nash , Jr. Duke Math. J. 81 (1995), no. 1, 227–250. 31, 46

    work page 1995

  43. [43]

    Kenig, J

    C. Kenig, J. Pipher, The Dirichlet problem for elliptic equations with drift terms , Publ. Mat. 45 (2001), no. 1, 199–217. 4

    work page 2001

  44. [44]

    Lieberman, H¨ older continuity of the gradient of solutions of uniformly parabolic equations with conor- mal boundary conditions, Ann

    G. Lieberman, H¨ older continuity of the gradient of solutions of uniformly parabolic equations with conor- mal boundary conditions, Ann. Mat. Pura Appl. (4) 148 (1987), 77–99. 17, 37

    work page 1987

  45. [45]

    Mitrea, The initial Dirichlet boundary value problem for general second order parabolic systems in nonsmooth manifolds, Comm

    M. Mitrea, The initial Dirichlet boundary value problem for general second order parabolic systems in nonsmooth manifolds, Comm. Partial Differential Equations 26 (2001), no. 11–12, 1975–2036. 5, 13

    work page 2001

  46. [46]

    Neˇ cas,Les methodes directes en theorie des equations elliptiques (French) , Masson et Cie, Editeurs, Paris; Academia, Editeurs, Prague (1967), 351 pp

    J. Neˇ cas,Les methodes directes en theorie des equations elliptiques (French) , Masson et Cie, Editeurs, Paris; Academia, Editeurs, Prague (1967), 351 pp. 2, 96

    work page 1967

  47. [47]

    Nystr¨ om,The Dirichlet problem for second order parabolic operators, Indiana Univ

    K. Nystr¨ om,The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J. 46 no.1 (1997), 183–245. 14, 15

    work page 1997

  48. [48]

    Nystr¨ om,Boundary Value Problems and Duality between Lp Dirichlet and Regularity Problems for Second Order Parabolic Systems in Non-Cylindrical Domains , Collect

    K. Nystr¨ om,Boundary Value Problems and Duality between Lp Dirichlet and Regularity Problems for Second Order Parabolic Systems in Non-Cylindrical Domains , Collect. Math.57(2006), no.1, 93–119. 5, 7, 13

    work page 2006

  49. [49]

    Mourgoglou, B

    M. Mourgoglou, B. Poggi, X. Tolsa, Solvability of the Poisson-Dirichlet problem with interior data in Lp′ -Carleson spaces and its applications to the Lp-regularity problem, arXiv:2207.10554, to appear in J. Eur. Math. Sci. 2, 4

    work page arXiv

  50. [50]

    Rivera-Noriega, Some remarks on certain parabolic differential operators over non-cylindrical do- mains, ProQuest LLC, Ann Arbor, MI, 2001, pp

    J. Rivera-Noriega, Some remarks on certain parabolic differential operators over non-cylindrical do- mains, ProQuest LLC, Ann Arbor, MI, 2001, pp. 110, ISBN: 978-0493-37327-0 73

    work page 2001

  51. [51]

    Rivera-Noriega, Absolute continuity of parabolic measure and area integral estimates in non-cylindrical domains, Ind

    J. Rivera-Noriega, Absolute continuity of parabolic measure and area integral estimates in non-cylindrical domains, Ind. Univ. Math. J. 52 (2003), no. 2, 477–525. 74

    work page 2003

  52. [52]

    Shen, Extrapolation for the Lp Dirichlet problem in Lipschitz domains , Acta Math

    Z. Shen, Extrapolation for the Lp Dirichlet problem in Lipschitz domains , Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 6, 1074–1084. 4, 14 112 DINDO ˇS, LI, AND PIPHER

    work page 2019

  53. [53]

    Ulmer, Perturbation theory for the parabolic regularity problem , Preprint arXiv:2408.12529

    M. Ulmer, Perturbation theory for the parabolic regularity problem , Preprint arXiv:2408.12529. 5, 16, 46

    work page arXiv

  54. [54]

    Ulmer, Lp Boundary Value Problems for Elliptic and Parabolic Operators , University of Edinburgh, thesis (2024)

    M. Ulmer, Lp Boundary Value Problems for Elliptic and Parabolic Operators , University of Edinburgh, thesis (2024). 46 School of Mathematics, The University of Edinburgh and Maxwell Institute of Mathe- matical Sciences, Edinburgh, UK Email address : M.Dindos@ed.ac.uk School of Mathematics, The University of Edinburgh and Maxwell Institute of Mathe- matica...

    work page 2024