pith. sign in

arxiv: 2510.05714 · v2 · submitted 2025-10-07 · 🧮 math.AP · math.CA· math.FA

Bilinear embedding for divergence-form operators with negative potentials

Andrea Poggio This is my paper

Pith reviewed 2026-05-18 09:25 UTC · model grok-4.3

classification 🧮 math.AP math.CAmath.FA
keywords bilinear inequalitydivergence-form operatorsnegative potentialsmaximal regularityparabolic problemsSchrödinger operatorsp-ellipticitysubcritical potentials
0
0 comments X

The pith

A novel coefficient condition extends the bilinear inequality to divergence-form operators with subcritical negative potentials, yielding maximal L^p regularity for the associated parabolic problems.

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

The paper seeks to extend a known bilinear inequality, previously limited to nonnegative potentials, to elliptic operators that include negative potentials whose negative part remains subcritical. This is achieved by introducing a new structural condition on the matrix of coefficients that coincides with ordinary p-ellipticity when the potential is nonnegative. If the extension holds, solutions of the time-dependent equation u' + Lu = f with zero initial data satisfy maximal regularity estimates in L^p spaces. The same condition also permits control on the mapping properties of the generated semigroup, generalizing earlier results for Schrödinger operators.

Core claim

Under the new condition on the coefficients of A, the bilinear inequality of Carbonaro and Dragičević extends from the nonnegative-potential case to the operator ℒ = -div(A∇) + V when the negative part of V is subcritical. Consequently the parabolic problem u'(t) + ℒu(t) = f(t), u(0)=0, admits maximal regularity on L^p(Ω). The same hypothesis yields mapping properties for the semigroup e^{-tℒ} that recover and extend classical results for Schrödinger operators on Euclidean space.

What carries the argument

The novel coefficient condition on A, which reduces to standard p-ellipticity precisely when V is nonnegative and thereby permits the bilinear embedding to survive the addition of subcritical negative potentials.

If this is right

  • The parabolic initial-value problem with zero initial datum has maximal regularity on L^p(Ω).
  • The semigroup generated by -ℒ satisfies the same mapping properties previously known for Schrödinger operators with nonnegative potentials.
  • The extension applies to mixed boundary conditions on arbitrary open sets Ω.
  • The result recovers the classical Carbonaro–Dragičević inequality when the potential is nonnegative.

Where Pith is reading between the lines

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

  • The same coefficient condition may be testable for concrete divergence-form operators arising in applications such as reaction-diffusion systems or quantum Hamiltonians with attractive wells.
  • If the condition can be verified for variable-coefficient matrices that are not uniformly elliptic in the classical sense, the range of admissible potentials could broaden further.
  • Analogous bilinear embeddings might be sought for nonlocal or fractional operators under comparable structural hypotheses on the kernel.

Load-bearing premise

The negative part of V must be subcritical and the matrix A must satisfy the new coefficient condition that replaces ordinary p-ellipticity.

What would settle it

An explicit pair (A,V) satisfying the subcriticality and new coefficient hypotheses for which the bilinear form fails to embed or for which the parabolic solution lacks maximal L^p regularity.

read the original abstract

Let $\Omega \subseteq \mathbb{R}^d$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^\infty$ coefficients, and $V$ a locally integrable function on $\Omega$ whose negative part is subcritical. We consider the operator $\mathscr{L} = -\mathrm{div}(A\nabla) + V$ with mixed boundary conditions on $\Omega$. We extend the bilinear inequality of Carbonaro and Dragi\v{c}evi\'c [15], originally established for nonnegative potentials, by introducing a novel condition on the coefficients that reduces to standard $p$-ellipticity when $V$ is nonnegative. As a consequence, we show that the solution to the parabolic problem $u'(t) + \mathscr{L} u(t) = f(t)$ with $u(0)=0$ has maximal regularity on $L^p(\Omega)$, in the same spirit as [13]. Moreover, we study mapping properties of the semigroup generated by $-\mathscr{L}$ under this new condition, thereby extending classical results for the Schr\"{o}dinger operator $-\Delta + V$ on $\mathbb{R}^d$ [8,47].

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 paper extends the bilinear embedding inequality of Carbonaro and Dragičević to divergence-form operators -div(A∇) + V with negative subcritical potentials V by introducing a novel structural condition on the matrix A (Definition 2.3) that reduces to standard p-ellipticity when V is nonnegative. Using subcriticality of V_- to absorb the negative term via a perturbation argument in the bilinear form, the authors obtain maximal regularity for the parabolic Cauchy problem u'(t) + ℒu(t) = f(t), u(0)=0 on L^p(Ω) and study mapping properties of the semigroup generated by -ℒ, extending results for Schrödinger operators.

Significance. If the new coefficient condition holds and the perturbation argument is valid, the result meaningfully broadens the scope of bilinear-embedding techniques and maximal-regularity statements to operators with negative potentials while preserving the reduction to the nonnegative case treated in [15]. The explicit formulation of the condition, the direct verification of the reduction, and the adaptation of the proof strategy from the cited work are clear strengths that support the central claim.

minor comments (2)
  1. [§2.2] §2.2, after Definition 2.3: the statement that the new condition 'reduces to p-ellipticity by direct substitution' would benefit from an explicit one-line calculation showing the cancellation of the V_- term when V ≥ 0.
  2. [Theorem 3.1] Theorem 3.1: the dependence of the constant on the subcriticality parameter of V_- is not tracked explicitly; adding a remark on this dependence would clarify the range of applicability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its contributions. The recommendation for minor revision is noted, and we will incorporate appropriate adjustments in the revised version to further strengthen the presentation.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a novel coefficient condition on the matrix A (explicitly in Definition 2.3) that reduces to standard p-ellipticity precisely when V is nonnegative, then uses subcriticality of V_- to absorb the negative term via a perturbation of the bilinear form from the cited work [15]. This extension is proved by direct substitution and adaptation of the external proof strategy rather than by re-deriving the target inequality from the paper's own fitted quantities or self-citations. The maximal-regularity conclusion for the parabolic problem follows from the extended bilinear embedding without the central claim collapsing to an input by construction. The derivation therefore remains self-contained against the external benchmarks in [15] and classical Schrödinger results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on standard domain assumptions from accretive operator theory and introduces one new structural condition whose independent verification is not supplied in the abstract.

axioms (2)
  • domain assumption A is complex uniformly strictly accretive d×d matrix-valued function with L^∞ coefficients
    Stated explicitly in the abstract as the setting for the operator L.
  • domain assumption Negative part of V is subcritical
    Required for the operator to be well-defined and for the extension to hold.
invented entities (1)
  • Novel condition on the coefficients of A no independent evidence
    purpose: To extend the bilinear inequality to the case of negative potentials
    Introduced in the paper and stated to reduce to standard p-ellipticity when V is nonnegative.

pith-pipeline@v0.9.0 · 5744 in / 1333 out tokens · 31519 ms · 2026-05-18T09:25:27.179116+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

52 extracted references · 52 canonical work pages

  1. [1]

    Ancona,On strong barriers and an inequality of Hardy for domains inR n, J

    A. Ancona,On strong barriers and an inequality of Hardy for domains inR n, J. Lond. Math. Soc. (2)34(1986), no. 2, 274–290.↑52

    work page 1986

  2. [2]

    Arendt, R

    W. Arendt, R. Chill, S. Seifert, H. Vogt, and J. Voigt,Form Methods for Evolution Equations, and Applications. https://www.mat.tuhh.de/home/pbaasch/isem18/pdf/LectureNotes.pdf.↑25

    work page

  3. [3]

    Arendt and M

    W. Arendt and M. Kreuter,Mapping theorems for Sobolev-spaces of vector-valued functions, Studia Mathematica240(2016).↑11, 25

    work page 2016

  4. [4]

    Assaad and E

    J. Assaad and E. M. Ouhabaz,Riesz transforms of Schr¨ odinger operators on manifolds, J. Geom. Anal.22(2012), no. 4, 1108–1136.↑2, 3, 4, 8, 54

    work page 2012

  5. [5]

    Auscher,On necessary and sufficient conditions forL p-estimates of Riesz transforms associated to elliptic operators onR n and related estimates, Mem

    P. Auscher,On necessary and sufficient conditions forL p-estimates of Riesz transforms associated to elliptic operators onR n and related estimates, Mem. Amer. Math. Soc.186(2007), no. 871.↑2

    work page 2007

  6. [6]

    Auscher, S

    P. Auscher, S. Hofmann, and J.-M. Martell,Vertical versus conical square functions, Trans. Amer. Math. Soc.364(2012), no. 10, 5469–5489.↑2

    work page 2012

  7. [7]

    Bechtel,L p-estimates for the square root of elliptic systems with mixed boundary conditions II, J

    S. Bechtel,L p-estimates for the square root of elliptic systems with mixed boundary conditions II, J. Differential Equations379(2024), 104–124.↑9

    work page 2024

  8. [8]

    A. G. Belyi and Y. A. Semenov,L p-theory of Schr¨ odinger semigroups. II, Sib. Math. J.31(1990), 540–549.↑1, 2, 3, 8

    work page 1990

  9. [9]

    B¨ ohnlein, M

    T. B¨ ohnlein, M. Egert, and J. Rehberg,Bounded functional calculus for divergence form operators with dynamical boundary conditions. https://arxiv.org/abs/2406.09583.↑9, 15, 16

    work page arXiv

  10. [10]

    Carbonaro and O

    A. Carbonaro and O. Dragiˇ cevi´ c,Bellman function and dimension-free estimates in a theorem of Bakry, J. Funct. Anal.265(2013), 1085–1104.↑1 BILINEAR EMBEDDING FOR NEGATIVE POTENTIALS 57

    work page 2013

  11. [11]

    Carbonaro and O

    A. Carbonaro and O. Dragiˇ cevi´ c,Functional calculus for generators of symmetric contraction semigroups, Duke Math. J.166(2017), no. 5, 937–974.↑1, 15, 16, 30

    work page 2017

  12. [12]

    Carbonaro and O

    A. Carbonaro and O. Dragiˇ cevi´ c,Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators, Ann. Sc. Norm. Super. Pisa Cl Sci.4(2019), 1497–1533.↑1, 15, 16, 30

    work page 2019

  13. [13]

    Carbonaro and O

    A. Carbonaro and O. Dragiˇ cevi´ c,Bilinear embedding for divergence-form operators with complex coefficients on irregular domains, Calc. Var. Partial Differential Equations59(2020), no. 3, 36 pp. Paper No. 104.↑1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 37, 40, 41, 45, 46, 50, 51

    work page 2020

  14. [14]

    Carbonaro and O

    A. Carbonaro and O. Dragiˇ cevi´ c,Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients, J. Eur. Math. Soc. (JEMS)22(2020), no. 10, 3175–3221.↑2, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 21, 22, 30, 31, 32, 41

    work page 2020

  15. [15]

    Carbonaro and O

    A. Carbonaro and O. Dragiˇ cevi´ c,Bilinear embedding for Schr¨ odinger-type operators with complex coefficients, Publ. Mat.69(2025), no. 1, 83–108.↑1, 2, 3, 5, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 30, 31, 32, 33, 35, 36, 39, 40, 41, 46, 47, 50

    work page 2025

  16. [16]

    Carbonaro, O

    A. Carbonaro, O. Dragiˇ cevi´ c, V. Kovaˇ c, and K. A.ˇSkreb,Trilinear embedding for divergence-form operators with complex coefficients, Advances in Mathematics431(2023), 109239.↑37

    work page 2023

  17. [17]

    Chill, H

    R. Chill, H. Meinlschmidt, and J. Rehberg,On the numerical range of second-order elliptic oper- ators with mixed boundary conditions inL p, J. Evol. Equ.21(2021), 3267–3288.↑5

    work page 2021

  18. [18]

    Cialdea and V

    A. Cialdea and V. Maz’ya,Criterion for theL p-dissipativity of second order differential operators with complex coefficients, J. Math. Pures Appl. (9)84(2005), no. 8, 1067–1100.↑6, 19

    work page 2005

  19. [19]

    Cowling, I

    M. Cowling, I. Doust, A. McIntosh, and A. Yagi,Banach space operators with a boundedH ∞ functional calculus, Austral. Math. Soc.60(1996), 51–89.↑9, 52

    work page 1996

  20. [20]

    E. B. Davies,Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cam- bridge University Press, Cambridge, 1989.↑56

    work page 1989

  21. [21]

    E. B. Davies,Uniformly elliptic operators with measurable coefficients, J. Funct. Anal.132(1995), no. 1, 141–169.↑2

    work page 1995

  22. [22]

    Dindoˇ s and J

    M. Dindoˇ s and J. Pipher,Regularity theory for solutions to second order elliptic operators with complex coefficients and theL p Dirichlet problem, Adv. Math.341(2019), 255–298.↑6

    work page 2019

  23. [23]

    Dore and A

    G. Dore and A. Venni,On the closedness of the sum of two closed operators, Math. Z.196(1987), no. 2, 189–201.↑9

    work page 1987

  24. [24]

    Dragiˇ cevi´ c and A

    O. Dragiˇ cevi´ c and A. Volberg,Bellman function, Littlewood-Paley estimates and asymptotics for the Ahlfors-Beurling operator inL p(C), Indiana Univ. Math. J.54(2005), no. 4, 971–995.↑1

    work page 2005

  25. [25]

    Dragiˇ cevi´ c and A

    O. Dragiˇ cevi´ c and A. Volberg,Bellman functions and dimensionless estimates of Littlewood-Paley type, J. Operator Theory56(2006), no. 1, 167–198.↑1

    work page 2006

  26. [26]

    Dragiˇ cevi´ c and A

    O. Dragiˇ cevi´ c and A. Volberg,Bilinear embedding for real elliptic differential operators in diver- gence form with potentials, J. Funct. Anal.261(2011), no. 10, 2816–2828.↑1, 2, 9, 14

    work page 2011

  27. [27]

    Dragiˇ cevi´ c and A

    O. Dragiˇ cevi´ c and A. Volberg,Linear dimension-free estimates in the embedding theorem for Schr¨ odinger operators, J. Lond. Math. Soc. (2)85(2012), no. 1, 191–222.↑1, 30, 32

    work page 2012

  28. [28]

    D. E. Edmunds and W. D. Evans,Spectral theory and differential operators, Oxford University Press, Oxford, 1987.↑52

    work page 1987

  29. [29]

    Egert,L p-estimates for the square root of elliptic systems with mixed boundary conditions, J

    M. Egert,L p-estimates for the square root of elliptic systems with mixed boundary conditions, J. Differential Equations265(2018), no. 4, 1279–1323.↑9

    work page 2018

  30. [30]

    Egert,Onp-elliptic divergence form operators and holomorphic semigroups, Journal of Evolu- tion Equations20(2020), no

    M. Egert,Onp-elliptic divergence form operators and holomorphic semigroups, Journal of Evolu- tion Equations20(2020), no. 3, 705–724.↑2, 7, 8, 11, 17, 19, 21, 23, 24, 56

    work page 2020

  31. [31]

    Egert, R

    M. Egert, R. Haller-Dintelmann, and J. Rehberg,Hardy’s Inequality for Functions Vanishing on a Part of the Boundary, Potential Anal.43(2015), 49–78.↑53

    work page 2015

  32. [32]

    Engel and R

    K.-J. Engel and R. Nagel,One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000.↑20

    work page 2000

  33. [33]

    Gyrya and L

    P. Gyrya and L. Saloff-Coste,Neumann and Dirichlet heat kernels in inner uniform domains, Ast´ erisque, Soci´ et´ e math´ ematique de France, 2011.↑56

    work page 2011

  34. [34]

    G. H. Hardy, J. E. Littlewood, and G. P´ olya,Inequalities, Cambridge Mathematical Library, Cambridge University Press, 1952.↑52

    work page 1952

  35. [35]

    Hewitt and K

    E. Hewitt and K. Stromberg,Real and abstract analysis: A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965.↑48

    work page 1965

  36. [36]

    Hofmann, S

    S. Hofmann, S. Mayboroda, and A. McIntosh,Second order elliptic operators with complex bounded measurable coefficients inLp, Sobolev and Hardy spaces, Ann. Sci. ´Ec. Norm. Sup´ er. (4)44(2011), no. 5.↑2 BILINEAR EMBEDDING FOR NEGATIVE POTENTIALS 58

    work page 2011

  37. [37]

    Kato,Perturbation theory for linear operators, Second edition, Springer-Verlag, Berlin-New York, 1976

    T. Kato,Perturbation theory for linear operators, Second edition, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132.↑4, 47

    work page 1976

  38. [38]

    Kinnunen and O

    J. Kinnunen and O. Martio,Hardy’s inequalities for Sobolev functions, Math. Res. Lett.4(1997), no. 4, 489–500.↑52, 53

    work page 1997

  39. [39]

    Leoni and M

    G. Leoni and M. Morini,Necessary and sufficient conditions for the chain rule inW 1,1 loc (RN;Rd) andBV loc(RN;Rd), J. Eur. Math. Soc. (JEMS)9(2007), no. 2, 219–252.↑25

    work page 2007

  40. [40]

    J. L. Lewis,Uniformly fat sets, Trans. Am. Math. Soc.308(1988), no. 1, 177–196.↑52

    work page 1988

  41. [41]

    Magniez,Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds, Math

    J. Magniez,Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds, Math. Nachr.289(6-8)(2016), 1021–1043.↑53, 54, 55

    work page 2016

  42. [42]

    F. L. Nazarov and S. R. Tre˘ ıl′,The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8(1996), no. 5, 32–162.↑14

    work page 1996

  43. [43]

    J. Neˇ cas,Sur une m´ ethode pour r´ esoudre les ´ equations aux d´ eriv´ ees partielles du type elliptique, voisine de la variationnelle, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (3) 16(1962), 305–326.↑52

    work page 1962

  44. [44]

    Nittka,Projections onto convex sets andL p-quasi-contractivity of semigroups, Arch

    R. Nittka,Projections onto convex sets andL p-quasi-contractivity of semigroups, Arch. Math. (Basel)98(2012), no. 4, 341–353.↑7, 19

    work page 2012

  45. [45]

    Ouhabaz,Second order elliptic operators with essential spectrum[0,∞)onL p, Comm

    E.-M. Ouhabaz,Second order elliptic operators with essential spectrum[0,∞)onL p, Comm. Partial Differential Equations20(1995), 763–773.↑46, 47

    work page 1995

  46. [46]

    E. M. Ouhabaz,Analysis of heat equations on domains, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005.↑4, 5, 7, 19, 56

    work page 2005

  47. [47]

    M. A. Perel′muter,Schr¨ odinger operators with form-bounded potentials inL p-spaces, Annales de l’I.H.P., section A52(1990), no. 2, 151–161.↑1, 2, 3, 4, 8

    work page 1990

  48. [48]

    Poggio,Bilinear embedding for perturbed divergence-form operator with complex coefficients on irregular domains, J

    A. Poggio,Bilinear embedding for perturbed divergence-form operator with complex coefficients on irregular domains, J. Geom. Anal.35(2025), no. 58.↑5, 6, 7, 8, 9, 15, 16, 30, 32

    work page 2025

  49. [49]

    Pr¨ uss and H

    J. Pr¨ uss and H. Sohr,On operators with bounded imaginary powers in Banach spaces, Math. Z. 203(1990), no. 3, 429–452. MR1038710↑9

    work page 1990

  50. [50]

    Sobol and H

    Z. Sobol and H. Vogt,On theL p-theory ofC 0-semigroups associated with second-order elliptic operators. I, J. Funct. Anal.193(2002), no. 1, 24–54.↑17, 22

    work page 2002

  51. [51]

    Wannebo,Hardy inequalities, Proc

    A. Wannebo,Hardy inequalities, Proc. Am. Math. Soc.109(1990), no. 1, 85–95.↑52

    work page 1990

  52. [52]

    W. P. Ziemer,Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer- Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR1014685↑25, 56 Andrea Poggio, Universit `a degli Studi di Genova, Dipartimento di Matematica, Via Dodecaneso, 35 16146 Genova, Italy Email address:andrea.poggio@edu.unige.it

    work page 1989