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arxiv: 1907.03689 · v1 · pith:C4NBN6MZnew · submitted 2019-07-08 · 🧮 math.CV

On Competing Definitions for the Diederich-Forn{ae}ss Index

Phillip S. Harrington This is my paper

Pith reviewed 2026-05-25 00:49 UTC · model grok-4.3

classification 🧮 math.CV
keywords Diederich-Fornæss indexplurisubharmonic functionspseudoconvex domainsdefining functionsupper semi-continuous functionsLipschitz functionsboundary regularity
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The pith

The Diederich-Fornæss index is the same whether defined using upper semi-continuous or Lipschitz functions, and the same for C^k and C^2 functions when the boundary is C^k.

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

The paper proves two main equivalences for the Diederich-Fornæss index on bounded pseudoconvex domains in C^n. First, the index obtained from upper semi-continuous defining functions matches the one from Lipschitz functions. Second, when the boundary has C^k regularity for k at least 2, using C^k functions yields the same index as using C^2 functions. This matters to readers interested in complex geometry because it shows the index is stable across these classes of functions, allowing flexibility in how one constructs the defining functions while preserving the value of the index.

Core claim

Computing the Diederich-Fornæss index with respect to the family of upper semi-continuous functions is the same as computing it with respect to the family of Lipschitz functions. When the boundary of Ω is C^k, k≥2, the index with respect to the family of C^k functions is the same as the index with respect to the family of C^2 functions.

What carries the argument

The Diederich-Fornæss index defined relative to a family of functions as the supremum of exponents η where a function from the family satisfies the distance comparability and plurisubharmonicity conditions.

Load-bearing premise

Ω is a bounded pseudoconvex domain in C^n.

What would settle it

Constructing a bounded pseudoconvex domain where the largest η for Lipschitz functions is strictly smaller than for upper semi-continuous functions would falsify the first claim.

read the original abstract

Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex domain. We define the Diederich-Forn{\ae}ss index with respect to a family of functions to be the supremum over the set of all exponents $0<\eta<1$ such that there exists a function $\rho_\eta$ in this family such that $-\rho_\eta$ is comparable to the distance to the boundary of $\Omega$ on $\Omega$ and such that $-(-\rho_\eta)^\eta$ is plurisubharmonic on $\Omega$. We first prove that computing the Diederich-Forn{\ae}ss index with respect to the family of upper semi-continuous functions is the same as computing the Diederich-Forn{\ae}ss index with respect to the family of Lipschitz functions. When the boundary of $\Omega$ is $C^k$, $k\geq 2$, we prove that the Diederich-Forn{\ae}ss index with respect to the family of $C^k$ functions is the same as the Diederich-Forn{\ae}ss index with respect to the family of $C^2$ 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

0 major / 3 minor

Summary. The paper defines the Diederich-Fornæss index of a bounded pseudoconvex domain Ω ⊂ ℂ^n relative to a family of functions as the supremum of η ∈ (0,1) such that there exists ρ_η in the family with -ρ_η comparable to dist(·,∂Ω) and -(−ρ_η)^η plurisubharmonic on Ω. It proves that the index computed over upper semi-continuous functions equals the index over Lipschitz functions, and that when ∂Ω is C^k (k≥2) the index over C^k functions equals the index over C^2 functions.

Significance. If the equivalences hold, the results establish robustness of the Diederich-Fornæss index under changes of function class, showing that the value is insensitive to regularity beyond Lipschitz (or C^2 when the boundary is smoother). This could simplify both theoretical statements and explicit computations in the literature on plurisubharmonic defining functions.

minor comments (3)
  1. The abstract states the two main theorems but does not indicate the section numbers in which the proofs appear; adding explicit cross-references would improve readability.
  2. Notation for the index (e.g., whether it is denoted DF(Ω) or DF_F(Ω) for a family F) should be introduced once in the introduction and used consistently thereafter.
  3. The comparability -ρ_η ∼ dist(·,∂Ω) is invoked without an explicit constant; stating whether the constant may depend on η would clarify the definition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the paper and the recommendation of minor revision. The report lists no specific major comments.

Circularity Check

0 steps flagged

No significant circularity; equivalences proven from independent definitions

full rationale

The paper defines the Diederich-Fornæss index separately for each function family (usc, Lipschitz, C^k) as the sup of admissible η where a defining function ρ_η in that class satisfies the distance comparability and plurisubharmonicity of -(-ρ_η)^η. It then proves the indices coincide by showing that existence in the larger class implies existence in the smaller class (and conversely) via approximation/regularization that preserves the two required properties. These steps rely on the standard bounded pseudoconvex domain assumption and classical several-complex-variables tools; no step reduces by construction to a fitted input, self-citation, or renamed ansatz. The central claims are therefore non-circular equivalences between independently defined quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard background in several complex variables (plurisubharmonicity, pseudoconvexity) with no free parameters, invented entities, or ad-hoc axioms visible from the abstract.

axioms (1)
  • standard math Standard properties of plurisubharmonic functions and pseudoconvex domains in C^n
    Invoked in the definition of the index and the statements about comparability to distance to the boundary.

pith-pipeline@v0.9.0 · 5734 in / 1099 out tokens · 20812 ms · 2026-05-25T00:49:00.995748+00:00 · methodology

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 3 internal anchors

  1. [1]

    Hartogs Domains and the Diederich Forn{\ae}ss Index

    M. Abdulsahib and P. S. Harrington, Hartogs domains and the diederich-fornæss index , arXiv:1809.02662, 2019

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  2. [2]

    D. E. Barrett, Behavior of the Bergman projection on the Diederich-Fornæs s worm , Acta Math. 168 (1992), 1–10

    work page 1992

  3. [3]

    Berndtsson and P

    B. Berndtsson and P. Charpentier, A Sobolev mapping property of the Bergman kernel , Math. Z. 235 (2000), 1–10. 20 PHILLIP S. HARRINGTON

    work page 2000

  4. [4]

    H. P. Boas and E. J. Straube, de Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the ∂-Neumann problem , J. Geom. Anal. 3 (1993), 225–235

    work page 1993

  5. [5]

    Chen and M.-C

    S.-C. Chen and M.-C. Shaw, Partial differential equations in several complex variable s, AMS/IP Studies in Advanced Mathematics, vol. 19, American M athematical Society, Provi- dence, RI, 2001

    work page 2001

  6. [6]

    Diederich and J

    K. Diederich and J. E. Fornaess, Pseudoconvex domains: an example with nontrivial Nebenh¨ ulle, Math. Ann. 225 (1977), 275–292

    work page 1977

  7. [7]

    , Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions , In- vent. Math. 39 (1977), 129–141

    work page 1977

  8. [8]

    Federer, Curvature measures, Trans

    H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491

    work page 1959

  9. [9]

    J. E. Fornaess and A.-K. Herbig, A note on plurisubharmonic defining functions in C2, Math. Z. 257 (2007), 769–781

    work page 2007

  10. [10]

    , A note on plurisubharmonic defining functions in Cn, Math. Ann. 342 (2008), 749– 772

    work page 2008

  11. [11]

    Harrington, The Diederich-Fornæss index and good vector fields , Indiana Univ

    P. Harrington, The Diederich-Fornæss index and good vector fields , Indiana Univ. Math. J. 68 (2019), 721–760

    work page 2019

  12. [12]

    Harrington and A

    P. Harrington and A. Raich, Defining functions for unbounded Cm domains, Rev. Mat. Iberoam. 29 (2013), 1405–1420

    work page 2013

  13. [13]

    P. S. Harrington, The order of plurisubharmonicity on pseudoconvex domains w ith Lipschitz boundaries, Math. Res. Lett. 15 (2008), 485–490

    work page 2008

  14. [14]

    , Global regularity for the ∂-Neumann operator and bounded plurisubharmonic exhaus- tion functions , Adv. Math. 228 (2011), 2522–2551

    work page 2011

  15. [15]

    Herbig and J

    A.-K. Herbig and J. D. McNeal, Convex defining functions for convex domains , J. Geom. Anal. 22 (2012), 433–454

    work page 2012

  16. [16]

    J. J. Kohn, Quantitative estimates for global regularity , Analysis and geometry in several com- plex variables (Katata, 1997), Trends Math., Birkh¨ auser B oston, Boston, MA, 1999, pp. 97– 128

    work page 1997

  17. [17]

    S. G. Krantz, B. Liu, and M. M. Peloso, Geometric analysis on the Diederich-Fornæss index , J. Korean Math. Soc. 55 (2018), 897–921

    work page 2018

  18. [18]

    S. G. Krantz and H. R. Parks, Distance to Ck hypersurfaces, J. Differential Equations 40 (1981), 116–120

    work page 1981

  19. [19]

    The Diederich-Forn{\ae}ss index I: for domains of non-trivial index

    B. Liu, The Diederich-Fornæss index I: for the non-trivial index , arXiv:1701.00293, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  20. [20]

    , The Diederich–Fornss index and the regularities on the ¯∂-Neumann problem , arXiv:1906.00315, 2019

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  21. [21]

    , The Diederich-Fornæss index II: For domains of trivial inde x, Adv. Math. 344 (2019), 289–310

    work page 2019

  22. [22]

    J. D. McNeal, A sufficient condition for compactness of the ∂-Neumann operator, J. Funct. Anal. 195 (2002), 190–205

    work page 2002

  23. [23]

    Pinton and G

    S. Pinton and G. Zampieri, The Diederich-Fornaess index and the global regularity of t he ¯∂-Neumann problem , Math. Z. 276 (2014), 93–113

    work page 2014

  24. [24]

    R. M. Range, A remark on bounded strictly plurisubharmonic exhaustion f unctions, Proc. Amer. Math. Soc. 81 (1981), 220–222

    work page 1981

  25. [25]

    Richberg, Stetige streng pseudokonvexe Funktionen , Math

    R. Richberg, Stetige streng pseudokonvexe Funktionen , Math. Ann. 175 (1968), 257–286

    work page 1968

  26. [26]

    E. J. Straube, Lectures on the L2-Sobolev theory of the ∂-Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (E MS), Z¨ urich, 2010

    work page 2010

  27. [27]

    B. M. W einstock, Some conditions for uniform H-convexity, Illinois J. Math. 19 (1975), 400–404. SCEN 309, 1 University of Arkansas, F ayetteville, AR 72701 E-mail address : psharrin@uark.edu

    work page 1975