pith. sign in

arxiv: 2604.14923 · v1 · submitted 2026-04-16 · 🧮 math.AP

Harnack inequality for mixed local-nonlocal weighted homogeneous equations

Nirjan Biswas , Stuti Das This is my paper

Pith reviewed 2026-05-10 10:15 UTC · model grok-4.3

classification 🧮 math.AP
keywords Harnack inequalitymixed local-nonlocal equationsp-Laplacianfractional p-Laplacianweighted equationsDe Giorgi-Nash-Moserweak solutions
0
0 comments X

The pith

Harnack inequality holds for weak solutions to mixed local-nonlocal equations with scaling-subcritical weights.

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

The paper establishes that weak solutions of the mixed equation combining the local p-Laplacian and its fractional counterpart satisfy the Harnack inequality when the right-hand side weight belongs to a scaling-subcritical Lebesgue space. Positive solutions are then bounded above and below by constants depending only on their average value over balls. The proof adapts the De Giorgi-Nash-Moser iteration scheme to absorb the local term, the nonlocal tail, and the weighted nonlinearity. A corresponding weak Harnack inequality is proved for supersolutions. The results extend earlier work on unweighted or special cases to general weights and to a broader class of integro-differential operators.

Core claim

We establish Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions to the mixed local-nonlocal equation −Δ_p u + (−Δ)_p^s u = V |u|^{p−2}u in Ω, where s ∈ (0,1), p ∈ (1,∞), and V lies in L^q(Ω) with q > d/p when d > p and q > 1 when d ≤ p.

What carries the argument

De Giorgi-Nash-Moser iteration together with expansion of positivity and tail estimates for the nonlocal term.

If this is right

  • The Harnack inequality applies to a wide class of integro-differential operators whose prototype is the fractional p-Laplacian.
  • Regularity results previously known only for unweighted or special weights now hold for general scaling-subcritical weights.
  • Both the strong Harnack inequality for solutions and the weak version for supersolutions follow from the same De Giorgi-Nash-Moser framework.

Where Pith is reading between the lines

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

  • The method may adapt to other mixed local-nonlocal operators whose scaling is comparable to the p-homogeneous case.
  • Sharpness of the weight integrability condition could be checked by constructing explicit singular solutions in the unit ball.

Load-bearing premise

The weight V must lie in the scaling-subcritical Lebesgue space L^q(Ω) so that the right-hand side can be absorbed into the iterative estimates without destroying the Harnack conclusion.

What would settle it

A weak solution or supersolution that violates the Harnack inequality when V is placed in L^q with q at or below the critical threshold.

read the original abstract

We consider the following class of mixed local-nonlocal equations: \begin{align}\label{abs}\tag{$\mathcal{P}$} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{align} where $s \in (0,1), p \in (1, \infty)$, and the weight function $V$ lies in scaling subcritical Lebesgue space $L^q(\Omega)$ where $q>\frac{d}{p}$ when $d>p$ and $q>1$ when $d \le p$. We establish Harnack inequality for weak solution and weak Harnack inequality for weak supersolution to ($\mathcal{P}$). Our approach is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. Our results also apply to integro-differential operators, with the prototype given by $(-\Delta)_p^s$. This work generalizes some regularity results of Garain-Kinnunen (Trans. Am. Math. Soc., 375(8), 2022) and Garain (Nonlinear Anal., 256, 2025) to the setting of general weight functions.

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

2 major / 3 minor

Summary. The manuscript establishes the Harnack inequality for weak solutions and the weak Harnack inequality for weak supersolutions of the mixed local-nonlocal equation (P): −Δ_p u + (−Δ)_p^s u = V |u|^{p−2}u in Ω, where s ∈ (0,1), p ∈ (1,∞), and the weight V lies in the scaling-subcritical Lebesgue space L^q(Ω) with q > d/p when d > p and q > 1 when d ≤ p. The proof adapts the De Giorgi–Nash–Moser iteration, expansion of positivity, and tail estimates for the nonlocal term, and the results are stated to apply to general integro-differential operators with prototype (−Δ)_p^s. The work generalizes the unweighted results of Garain–Kinnunen (Trans. AMS 2022) and Garain (Nonlinear Anal. 2025).

Significance. If the estimates are verified, the result provides a natural extension of Harnack theory to mixed local-nonlocal operators with subcritical potentials, which is useful for applications involving inhomogeneous terms. The subcriticality condition on q is the standard threshold that permits absorption of the lower-order term via Hölder and Sobolev embeddings inside the Caccioppoli and logarithmic estimates, while the nonlocal tail is controlled separately by fractional Sobolev inequalities; this separation is a strength of the approach. The paper correctly credits the foundational tools and prior unweighted works.

major comments (2)
  1. [§4] §4 (Caccioppoli-type estimates): the absorption of the right-hand side term ∫ V |u|^p via Hölder’s inequality with the given q produces a remainder controlled by ||V||_q times a Sobolev norm of u; it must be confirmed that this remainder is absorbed without introducing a smallness restriction on ||V||_q or altering the iteration constants that enter the final Harnack constant.
  2. [§5] §5 (expansion of positivity): the tail term arising from the nonlocal operator is estimated separately, but the interaction between this tail and the weighted lower-order term V|u|^{p−2}u during the positivity expansion step is not explicitly quantified; a concrete bound showing that the tail remains controllable under the stated integrability of V is needed to close the argument.
minor comments (3)
  1. [Abstract] Abstract: the phrasing “Harnack inequality for weak solution and weak Harnack inequality for weak supersolution” should be corrected for grammar and parallelism.
  2. [Introduction] Introduction: the precise definition of weak solutions (including the integrability requirements imposed by V) should be stated before the main theorems, rather than deferred to a later section.
  3. [§1] Notation: the symbol for the mixed operator should be introduced consistently; the current use of (P) for the equation and the operator itself is slightly ambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. Below we address the major comments point by point, clarifying the absorption mechanism and the control of the tail term.

read point-by-point responses

  1. Referee: [§4] §4 (Caccioppoli-type estimates): the absorption of the right-hand side term ∫ V |u|^p via Hölder’s inequality with the given q produces a remainder controlled by ||V||_q times a Sobolev norm of u; it must be confirmed that this remainder is absorbed without introducing a smallness restriction on ||V||_q or altering the iteration constants that enter the final Harnack constant.

    Authors: The absorption is performed in the proof of the Caccioppoli inequality in Section 4. After applying Hölder's inequality with exponent q and its conjugate, the term ||V||_q ||u||_{L^{p q'}} is controlled by the Sobolev embedding since q is subcritical, i.e., the exponent p q' < p^* where p^* is the Sobolev conjugate. This allows absorption into the gradient term on the left-hand side with a constant that depends on ||V||_q but does not require it to be small. The Moser iteration constants are adjusted accordingly but remain independent of the solution, leading to a Harnack constant that depends on ||V||_q, d, p, s, which is the expected dependence. We will include an explicit remark in the revised manuscript to highlight this point. revision: partial

  2. Referee: [§5] §5 (expansion of positivity): the tail term arising from the nonlocal operator is estimated separately, but the interaction between this tail and the weighted lower-order term V|u|^{p−2}u during the positivity expansion step is not explicitly quantified; a concrete bound showing that the tail remains controllable under the stated integrability of V is needed to close the argument.

    Authors: During the expansion of positivity, the tail estimate for the nonlocal part is derived using the definition of the weak solution and fractional Sobolev inequalities, yielding a bound in terms of the L^p norm outside a ball, independent of V. The lower-order term involving V is treated in the local energy estimates prior to the positivity step, where it is absorbed as in the Caccioppoli inequality. To address the interaction explicitly, we will add a lemma or remark in Section 5 providing the bound: the contribution of the tail is estimated by C (tail)^{p} + epsilon times the local term, with the V term absorbed separately without affecting the tail control, thanks to the subcriticality. This closes the argument without additional assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external tools to new setting

full rationale

The paper derives Harnack inequalities for the mixed local-nonlocal equation by extending the standard De Giorgi-Nash-Moser iteration, Caccioppoli estimates, and expansion-of-positivity arguments to include the subcritical weight term V|u|^{p-2}u. The condition q > d/p (or q > 1) is the scaling threshold that permits absorption of the potential via Hölder and Sobolev embeddings inside those estimates; this is a direct application of known embedding theory rather than a fitted parameter or self-referential definition. The tail term from the nonlocal part is controlled by the usual fractional Sobolev inequality. Citations are to independent prior results on the unweighted mixed case (Garain-Kinnunen and Garain), which supply the base estimates without overlap in authorship or reduction to the present paper's inputs. No step in the claimed chain collapses to the paper's own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background results from nonlinear analysis and the applicability of De Giorgi-Nash-Moser theory to the mixed operator; no free parameters or new entities are introduced.

axioms (2)
  • standard math The p-Laplacian and fractional p-Laplacian satisfy the usual structural inequalities and comparison principles used in De Giorgi-Nash-Moser theory.
    Invoked as the foundation for the iteration and positivity expansion.
  • domain assumption Tail estimates and expansion of positivity hold for the mixed local-nonlocal operator under the given integrability of V.
    Central to absorbing the right-hand side term.

pith-pipeline@v0.9.0 · 5523 in / 1328 out tokens · 41532 ms · 2026-05-10T10:15:01.083995+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

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    C. A. Antonini and M. Cozzi. Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type.J. Differential Equations, 425:342–382, 2025. 2

    work page 2025

  2. [2]

    M. Biroli. Schrödinger type and relaxed Dirichlet problems for the subellipticp-Laplacian.Po- tential Anal., 15(1-2):1–16, 2001. ICPA98 (Hammamet). 2

    work page 2001

  3. [3]

    Biswas, M

    A. Biswas, M. Modasiya, and A. Sen. Boundary regularity of mixed local-nonlocal operators and its application.Ann. Mat. Pura Appl. (4), 202(2):679–710, 2023. 2

    work page 2023

  4. [4]

    Biswas and E

    A. Biswas and E. Topp. InteriorC1,α regularity of mixed local-nonlocal(p, q)-energy minimizers forp≤sq.arXiv:2512.07481, 2025. 2

    work page arXiv 2025

  5. [5]

    Biswas and E

    A. Biswas and E. Topp. Lipschitz regularity of fractionalp-Laplacian.Ann. PDE, 11(2):Paper No. 27, 43, 2025. 2

    work page 2025

  6. [6]

    Lipschitzpotentialestimatesfordiffusionwithjumps.NoDEA Nonlinear Differential Equations Appl., 32(5):Paper No

    N.BiswasandH.Prasad. Lipschitzpotentialestimatesfordiffusionwithjumps.NoDEA Nonlinear Differential Equations Appl., 32(5):Paper No. 88, 26, 2025. 2

    work page 2025

  7. [7]

    Brasco, E

    L. Brasco, E. Lindgren, and A. Schikorra. Higher Hölder regularity for the fractionalp-Laplacian in the superquadratic case.Adv. Math., 338:782–846, 2018. 2

    work page 2018

  8. [8]

    S.-S. Byun, D. Kumar, and H.-S. Lee. Global gradient estimates for the mixed local and nonlocal problems with measurable nonlinearities.Calc. Var. Partial Differential Equations, 63(2):Paper No. 27, 48, 2024. 2

    work page 2024

  9. [9]

    Chiarenza, E

    F. Chiarenza, E. Fabes, and N. Garofalo. Harnack’s inequality for Schrödinger operators and the continuity of solutions.Proc. Amer. Math. Soc., 98(3):415–425, 1986. 2

    work page 1986

  10. [10]

    De Filippis and G

    C. De Filippis and G. Mingione. Gradient regularity in mixed local and nonlocal problems. Mathematische Annalen, pages 1–68, 2022. 2

    work page 2022

  11. [11]

    Di Castro, T

    A. Di Castro, T. Kuusi, and G. Palatucci. Nonlocal Harnack inequalities.J. Funct. Anal., 267(6):1807–1836, 2014. 2, 3, 8, 9, 13, 15, 18

    work page 2014

  12. [12]

    Di Castro, T

    A. Di Castro, T. Kuusi, and G. Palatucci. Local behavior of fractionalp-minimizers.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33(5):1279–1299, 2016. 2

    work page 2016

  13. [13]

    Di Castro, T

    A. Di Castro, T. Kuusi, and G. Palatucci. Local behavior of fractionalp-minimizers.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33(5):1279–1299, 2016. 6, 11

    work page 2016

  14. [14]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012. 4

    work page 2012

  15. [15]

    DiBenedetto.Degenerate parabolic equations

    E. DiBenedetto.Degenerate parabolic equations. Universitext. Springer-Verlag, New York, 1993. 9

    work page 1993

  16. [16]

    M. Ding, Y. Fang, and C. Zhang. Local behavior of the mixed local and nonlocal problems with nonstandard growth.J. Lond. Math. Soc. (2), 109(6):Paper No. e12947, 34, 2024. 2 Harnack inequality for mixed local and nonlocalp-Laplace equations23

    work page 2024

  17. [17]

    L. C. Evans.Partial differential equations, volume 19 ofGraduate Studies in Mathematics. Amer- ican Mathematical Society, Providence, RI, second edition, 2010. 2

    work page 2010

  18. [18]

    P. Garain. Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations.Nonlinear Anal., 256:Paper No. 113796, 31, 2025. 3

    work page 2025

  19. [19]

    P. Garain. Two alternative proofs of weak harnack inequality for mixed local and nonlocalp-laplace equations with a nonhomogeneity.arXiv preprint arXiv:2510.04065, 2025. 2

    work page internal anchor Pith review arXiv 2025

  20. [20]

    Garain and J

    P. Garain and J. Kinnunen. On the regularity theory for mixed local and nonlocal quasilinear elliptic equations.Trans. Amer. Math. Soc., 375(8):5393–5423, 2022. 2, 3

    work page 2022

  21. [21]

    Garain and E

    P. Garain and E. Lindgren. Higher hölder regularity for mixed local and nonlocal degenerate elliptic equations.Calculus of Variations and Partial Differential Equations, 62(2):67, 2023. 2, 3

    work page 2023

  22. [22]

    Garain and E

    P. Garain and E. Lindgren. Higher Hölder regularity for the fractionalp-Laplace equation in the subquadratic case.Math. Ann., 390(4):5753–5792, 2024. 2

    work page 2024

  23. [23]

    Giaquinta and E

    M. Giaquinta and E. Giusti. On the regularity of the minima of variational integrals.Acta Math., 148:31–46, 1982. 19

    work page 1982

  24. [24]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order, volume 2. Springer, 1998. 21

    work page 1998

  25. [25]

    Giovagnoli, D

    D. Giovagnoli, D. Jesus, and L. Silvestre. C1+α regularity for fractionalp-harmonic functions,

    work page

  26. [26]

    Han and F

    Q. Han and F. Lin.Elliptic partial differential equations, volume 1 ofCourant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. 2

    work page 1997

  27. [27]

    John and L

    F. John and L. Nirenberg. On functions of bounded mean oscillation.Comm. Pure Appl. Math., 14:415–426, 1961. 2

    work page 1961

  28. [28]

    Kassmann

    M. Kassmann. A new formulation of Harnack’s inequality for nonlocal operators.C. R. Math. Acad. Sci. Paris, 349(11-12):637–640, 2011. 2

    work page 2011

  29. [29]

    Malý and W

    J. Malý and W. P. Ziemer.Fine regularity of solutions of elliptic partial differential equations, volume 51 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997. 2, 4

    work page 1997

  30. [30]

    Modasiya and A

    M. Modasiya and A. Sen. Fine boundary regularity for fully nonlinear mixed local-nonlocal prob- lems.J. Differential Equations, 452:Paper No. 113780, 47, 2026. 2

    work page 2026

  31. [31]

    J. Moser. On Harnack’s theorem for elliptic differential equations.Comm. Pure Appl. Math., 14:577–591, 1961. 1

    work page 1961

  32. [32]

    X. Su, E. Valdinoci, Y. Wei, and J. Zhang. Regularity results for solutions of mixed local and nonlocal elliptic equations.Math. Z., 302(3):1855–1878, 2022. 2

    work page 2022

  33. [33]

    X. Su, E. Valdinoci, Y. Wei, and J. Zhang. On some regularity properties of mixed local and nonlocal elliptic equations.J. Differential Equations, 416:576–613, 2025. 2

    work page 2025

  34. [34]

    N. S. Trudinger. On Harnack type inequalities and their application to quasilinear elliptic equa- tions.Comm. Pure Appl. Math., 20:721–747, 1967. 21 1Department of Mathematics, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pune 411008, India 2Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, ...

    work page 1967