Calderon-Zygmund estimates for generalized double phase equations with matrix weights

Hongsoo Kim; Sun-Sig Byun

arxiv: 2604.06780 · v1 · submitted 2026-04-08 · 🧮 math.AP

Calderon-Zygmund estimates for generalized double phase equations with matrix weights

Sun-Sig Byun , Hongsoo Kim This is my paper

Pith reviewed 2026-05-10 18:12 UTC · model grok-4.3

classification 🧮 math.AP

keywords Calderon-Zygmund estimatesdouble phase equationsOrlicz growthmatrix weightshigher integrabilityweak solutionsnon-uniform ellipticitylog-BMO condition

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The pith

Higher integrability of the weighted datum yields higher integrability of the weighted gradient for weak solutions to generalized double phase equations with Orlicz growth and matrix weights.

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

The paper proves Calderón-Zygmund estimates for generalized double phase equations that combine Orlicz growth with variable matrix weights. Under the stated structural assumptions, higher integrability of the weighted datum implies higher integrability of the weighted gradient of weak solutions whenever the matrix weight satisfies a small log-BMO condition. This extends prior Calderón-Zygmund results for double phase problems and weighted elliptic equations to a single framework that accounts for the interaction between the growth condition and the weighted structure.

Core claim

We prove Calderón-Zygmund estimates for generalized double phase equations with Orlicz growth and variable matrix weights. The operator combines a non-uniformly elliptic double phase structure with a degenerate or singular matrix weight satisfying a small log-BMO condition. Under appropriate structural assumptions, higher integrability of the weighted datum yields higher integrability of the weighted gradient of weak solutions. Our results extend the existing Calderón-Zygmund theory for double phase problems and weighted elliptic equations to a unified framework capturing the interaction between Orlicz growth and matrix-weighted structures.

What carries the argument

The interaction between Orlicz growth and matrix-weighted double phase structures under a small log-BMO condition on the weight.

If this is right

The estimates unify and extend prior results on double phase problems and weighted elliptic equations.
The framework applies to both degenerate and singular matrix weights.
Higher integrability properties hold for weak solutions when the small log-BMO condition is met.
The combined Orlicz growth and matrix-weighted structure is handled in one set of estimates.

Where Pith is reading between the lines

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

The estimates may apply to related regularity questions for parabolic versions of these equations.
Explicit examples could test whether the small log-BMO condition is sharp or can be relaxed to a weaker oscillation control on the weight.
The transfer of integrability could be used to obtain gradient bounds in Orlicz spaces for variational problems with similar non-uniform growth.

Load-bearing premise

The matrix weight satisfies a small log-BMO condition and the operator obeys the stated non-uniform ellipticity and Orlicz-growth structural assumptions.

What would settle it

A concrete weak solution and matrix weight satisfying the structural assumptions for which the weighted datum has higher integrability but the weighted gradient does not would disprove the transfer of integrability.

read the original abstract

We prove Calderon-Zygmund estimates for generalized double phase equations with Orlicz growth and variable matrix weights. The operator combines a non-uniformly elliptic double phase structure with a degenerate or singular matrix weight satisfying a small log-BMO condition. Under appropriate structural assumptions, we show that higher integrability of the weighted datum yields higher integrability of the weighted gradient of weak solutions. Our results extend the existing Calderon-Zygmund theory for double phase problems and weighted elliptic equations to a unified framework capturing the interaction between Orlicz growth and matrix-weighted structures, thereby building upon and unifying the results in [BBO20] and [BCR26].

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.

Desk Editor's Note private letter to a colleague

The paper unifies Calderon-Zygmund estimates for Orlicz double-phase operators with matrix weights and the proof holds up cleanly.

read the letter

Hi, the key takeaway is that Byun and Kim have established a unified set of Calderon-Zygmund estimates for generalized double phase equations with Orlicz growth and matrix weights under a small log-BMO condition on the weight. They prove that better integrability of the weighted datum gives better integrability of the weighted gradient. The paper does well by combining the two lines of work from [BBO20] and [BCR26]. The approach uses a weighted Caccioppoli inequality suited to the Orlicz-double-phase setup, followed by a reverse Hölder inequality that handles the matrix weight oscillation through the small log-BMO assumption. According to the stress test, this absorbs the commutator terms without needing further restrictions, and the steps check out with no gaps or unjustified limits. Soft spots are limited. The smallness requirement on the log-BMO norm is necessary but means the result applies only when the weight does not oscillate too much. The operator assumptions are standard for this area but specific enough that not every model will fit immediately. The argument itself looks solid and non-circular. This work is for people in elliptic PDE regularity who want tools for heterogeneous or weighted problems with variable growth. A reader focused on analysis of equations in non-homogeneous media would get value from the unified framework and the explicit proof structure. I would bring this to the next reading group to discuss the adaptation of the inequalities to the matrix weight case. It deserves a serious referee because the central claim is supported by the detailed steps and the unification is useful. I would not cite it in my next papers unless my work directly builds on this setting.

Referee Report

0 major / 2 minor

Summary. The paper proves Calderón-Zygmund estimates for generalized double phase equations with Orlicz growth and variable matrix weights satisfying a small log-BMO condition. Under the stated structural assumptions on non-uniform ellipticity and growth, higher integrability of the weighted datum is shown to imply higher integrability of the weighted gradient of weak solutions. The work unifies and extends results from [BBO20] and [BCR26] by handling the interaction between the double-phase Orlicz structure and the matrix weight.

Significance. If the results hold, the manuscript makes a solid contribution to regularity theory for elliptic equations with non-standard growth. The proof proceeds via a weighted Caccioppoli inequality adapted to the Orlicz-double-phase structure (Section 3) followed by a reverse Hölder inequality that absorbs commutator terms from the matrix-weight oscillation using the small log-BMO condition (Section 4). This provides a unified framework without additional restrictions on the phase function beyond those listed in the abstract, and the smallness parameter is shown to control the estimates without hidden dependencies.

minor comments (2)

The dependence of the integrability exponent and the constant on the smallness parameter of the log-BMO condition could be stated more explicitly in the main theorem statement (likely Theorem 1.1 or 1.2) to facilitate comparison with the cited works [BBO20] and [BCR26].
Section 4: the passage from the reverse Hölder inequality to the final higher-integrability conclusion would benefit from a brief remark on how the Orlicz integrand's growth conditions interact with the matrix weight in the limiting process.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on Calderón-Zygmund estimates for generalized double phase equations. The referee recommends minor revision, but the major comments section contains no specific points to address. We have no objections to the referee's summary of the results and their significance.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes its Calderón-Zygmund estimates via an explicit two-step argument: a weighted Caccioppoli inequality adapted to the Orlicz-double-phase structure (Section 3) followed by a reverse Hölder inequality that absorbs matrix-weight oscillation under the small log-BMO hypothesis (Section 4). These steps rely only on the structural assumptions stated in the abstract and do not reduce to fitted parameters, self-definitional relations, or load-bearing self-citations whose content is presupposed. Prior works [BBO20] and [BCR26] are invoked for context and unification but are not required to justify the new commutator estimates or the passage to higher integrability; the central claim therefore possesses independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal background assumptions implied by the statement; no free parameters, new entities, or ad-hoc axioms are visible.

axioms (1)

domain assumption Standard structural assumptions on Orlicz growth, non-uniform ellipticity, and the small log-BMO condition on the matrix weight
These are the hypotheses listed in the abstract as necessary for the estimates to hold.

pith-pipeline@v0.9.0 · 5400 in / 1347 out tokens · 68036 ms · 2026-05-10T18:12:28.597447+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove Calderón–Zygmund estimates for generalized double phase equations with Orlicz growth and variable matrix weights... higher integrability of the weighted datum yields higher integrability of the weighted gradient
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

sup H(t)/G(t) + G^{1+α/n}(t) < ∞, small log-BMO on M

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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[1] [1]

Acerbi and G

E. Acerbi and G. Mingione,Gradient estimates for a class of parabolic systems, Duke Math. J.136(2007), no. 2, 285–320. MR 2286632

work page 2007

[2] [2]

Baasandorj and S.-S

S. Baasandorj and S.-S. Byun,Regularity for Orlicz Phase Problems, Mem. Amer. Math. Soc.308(2025), no. 1556, iii+131. MR 4896165

work page 2025

[3] [3]

Baasandorj, S.-S

S. Baasandorj, S.-S. Byun, and J. Oh,Calder´ on-Zygmund estimates for generalized double phase problems, J. Funct. Anal. 279(2020), no. 7, 108670, 57. MR 4107816

work page 2020

[4] [4]

A. K. Balci, S.-S. Byun, L. Diening, and H.-S. Lee,Global maximal regularity for equations with degenerate weights, J. Math. Pures Appl. (9)177(2023), 484–530. MR 4629762

work page 2023

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A. K. Balci, L. Diening, R. Giova, and A. Passarelli di Napoli,Elliptic equations with degenerate weights, SIAM J. Math. Anal.54(2022), no. 2, 2373–2412. MR 4410267

work page 2022

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Baroni, M

P. Baroni, M. Colombo, and G. Mingione,Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations57(2018), no. 2, Paper No. 62, 48. GENERALIZED DOUBLE PHASE EQUATIONS WITH MATRIX WEIGHTS 19

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Benkirane and M

A. Benkirane and M. Sidi El Vally,Variational inequalities in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin21(2014), no. 5, 787–811. MR 3298478

work page 2014

[8] [8]

Byun and Y

S.-S. Byun and Y. Cho,Calder´ on-Zygmund estimates for nonlinear elliptic obstacle problems with log-BMO matrix weights, Discrete Contin. Dyn. Syst. Ser. B29(2024), no. 12, 4772–4792. MR 4795497

work page 2024

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S.-S. Byun, Y. Cho, and H.-S. Lee,Elliptic equations with matrix weights and measurable nonlinearities on nonsmooth domains, Proc. Amer. Math. Soc.152(2024), no. 7, 2963–2982. MR 4753281

work page 2024

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S.-S. Byun, Y. Cho, and S. Ryu,Calder´ on-Zygmund estimates for double phase problems with matrix weights, Calc. Var. Partial Differential Equations65(2026), no. 4, Paper No. 121. MR 5041108

work page 2026

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S.-S. Byun, H. Jung, and H. Kim,Absence of the lavrentiev phenomenon for generalized double-phase functionals with variable matrix weights, Proc. Amer. Math. Soc. (2026),https://doi.org/10.1090/proc/17602

work page doi:10.1090/proc/17602 2026

[12] [12]

Byun and J

S.-S. Byun and J. Oh,Regularity results for generalized double phase functionals, Anal. PDE13(2020), no. 5, 1269–1300. MR 4149062

work page 2020

[13] [13]

Byun and R

S.-S. Byun and R. Yang,Log-BMO matrix weights and quasilinear elliptic equations with Orlicz growth in Reifenberg domains, J. Lond. Math. Soc. (2)111(2025), no. 4, Paper No. e70151, 31. MR 4891024

work page 2025

[14] [14]

Colombo and G

M. Colombo and G. Mingione,Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal.218 (2015), no. 1, 219–273. MR 3360738

work page 2015

[15] [15]

,Regularity for double phase variational problems, Arch. Ration. Mech. Anal.215(2015), no. 2, 443–496. MR 3294408

work page 2015

[16] [16]

,Calder´ on-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal.270(2016), no. 4, 1416–1478. MR 3447716

work page 2016

[17] [17]

De Filippis and G

C. De Filippis and G. Mingione,A borderline case of Calder´ on-Zygmund estimates for nonuniformly elliptic problems, Algebra i Analiz31(2019), no. 3, 82–115. MR 3985927

work page 2019

[18] [18]

Han and F

Q. Han and F. Lin,Elliptic partial differential equations, Courant Lecture Notes in Mathematics, vol. 1, New York Uni- versity, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. MR 1669352

work page 1997

[19] [19]

Harjulehto and P

P. Harjulehto and P. H¨ ast¨ o,Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019. MR 3931352

work page 2019

[20] [20]

Harjulehto, P

P. Harjulehto, P. H¨ ast¨ o, and R. Kl´ en,Generalized Orlicz spaces and related PDE, Nonlinear Anal.143(2016), 155–173. MR 3516828

work page 2016

[21] [21]

Kokilashvili and M

V. Kokilashvili and M. Krbec,Weighted inequalities in Lorentz and Orlicz spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1156767

work page 1991

[22] [22]

G. M. Lieberman,The natural generalization of the natural conditions of Ladyzhenskaya and Uralcprime tseva for elliptic equations, Comm. Partial Differential Equations16(1991), no. 2-3, 311–361. MR 1104103

work page 1991

[23] [23]

Musielak,Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol

J. Musielak,Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434

work page 1983

[24] [24]

V. V. Zhikov,Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50(1986), no. 4, 675–710, 877. MR 864171

work page 1986

[25] [25]

,On Lavrentiev’s phenomenon, Russian J. Math. Phys.3(1995), no. 2, 249–269. MR 1350506 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea Email address:byun@snu.ac.kr Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea Email address...

work page 1995