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arxiv: 2604.23158 · v1 · submitted 2026-04-25 · 🧮 math.FA

Bourgain-Brezis spaces obtained by real interpolation

Eduard Curc\u{a} This is my paper

Pith reviewed 2026-05-08 07:09 UTC · model grok-4.3

classification 🧮 math.FA
keywords real interpolationdivergence operatorBourgain-Brezis equalityfunction spacesSobolev spacesinterpolation of solutionslinear equations
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The pith

If a function space X satisfies div(L^∞ ∩ X) = div X, then all its real interpolations with L^∞ do as well.

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

The paper establishes that the equality div(L^∞ ∩ X) = div X is preserved when X is replaced by any real interpolation space (L^∞, X)_{θ,q} for θ in (0,1) and q in [1,∞). This holds because a general method exists for interpolating solutions of linear equations such as the divergence equation. A sympathetic reader would care because the result automatically produces new families of spaces with the property, starting from any known example like certain Sobolev spaces, without deriving fresh estimates for each interpolated space. The approach thereby enlarges the collection of function spaces for which bounded representatives exist with prescribed divergence.

Core claim

Suppose X is a function space satisfying div(L^∞ ∩ X) = div X. Then the same equality holds for every real interpolation space X_{θ,q} = (L^∞, X)_{θ,q} with θ ∈ (0,1) and q ∈ [1,∞). The argument proceeds by applying a general interpolation technique that carries solutions of the linear divergence equation across the interpolation scale.

What carries the argument

The real interpolation functor (L^∞, X)_{θ,q} together with a general method that interpolates solutions of linear equations to preserve equality of images under the divergence operator.

If this is right

  • Every space known to satisfy the equality immediately yields an entire scale of interpolated spaces that also satisfy it.
  • The equality holds uniformly for all real interpolation parameters θ and q in the given ranges.
  • Starting from W^{1,d} on the torus, the spaces (L^∞, W^{1,d})_{θ,q} all satisfy div(L^∞ ∩ X_{θ,q}) = div X_{θ,q}.
  • The general interpolation method for linear equations extends the original equality to any space obtained this way.

Where Pith is reading between the lines

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

  • The same preservation might hold under other interpolation functors or for related first-order operators such as the curl.
  • These interpolated spaces could supply new function spaces in which to seek bounded solutions to divergence-form PDEs.
  • One verified space with the equality property generates a continuum of others, potentially streamlining arguments that rely on this image equality.

Load-bearing premise

The general method for interpolating solutions of linear equations applies directly to the divergence operator without extra conditions on X or the interpolation parameters.

What would settle it

Identify a concrete space X that satisfies div(L^∞ ∩ X) = div X, yet for some θ ∈ (0,1) and q ∈ [1,∞) there exists a vector field in X_{θ,q} whose divergence cannot be realized by any vector field in L^∞ ∩ X_{θ,q}.

read the original abstract

In 2002, Bourgain and Brezis proved that for the space $X=W^{1,d}$ (on $\mathbb{T}^{d}$, with $d\geq2$) we have the equality of images \begin{equation} \operatorname{div} (L^{\infty}\cap X)=\operatorname{div} X, \tag{$\ast$} \end{equation} i.e., given a vector field $v\in X$ there exists a vector field $u\in L^{\infty }\cap X$ such that $\operatorname{div} u=\operatorname{div] v $. In this paper we show that if $X$ is a function space satisfying ($\ast$) then, any real interpolation space $X_{\theta,q}=(L^{\infty},X)_{\theta,q}$ (where $\theta\in (0,1)$ and $q\in [1,\infty)$) also satisfies ($\ast$). The proof is based on a general method that allows us to interpolate solutions of linear equations.

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

1 major / 1 minor

Summary. The paper claims that if a function space X satisfies the Bourgain-Brezis equality div(L^∞ ∩ X) = div X, then every real interpolation space X_{θ,q} = (L^∞, X)_{θ,q} with θ ∈ (0,1) and q ∈ [1,∞) satisfies the same equality. The argument is based on a general method for interpolating solutions of linear equations applied to the divergence operator.

Significance. If the central claim holds, the result would systematically enlarge the class of spaces known to satisfy the Bourgain-Brezis property by including all real interpolants between L^∞ and any base space X that satisfies it. This could streamline applications in PDE theory and Sobolev embeddings, especially since the method is presented as general rather than case-by-case.

major comments (1)
  1. [Proof of the main result (following the abstract)] The manuscript invokes a general theorem on interpolating solutions of linear equations to transfer the range equality (*). However, (*) supplies only existential (nonlinear) preimages under div, while real interpolation functors preserve bounded linear operators. Without an explicit linear selection or bounded right inverse for div on L^∞ ∩ X, it is unclear how the general theorem applies directly to produce solutions in L^∞ ∩ X_{θ,q}. This step is load-bearing for the claim and requires a precise statement of the general theorem together with verification that its hypotheses are met by the divergence operator under (*).
minor comments (1)
  1. [Abstract] In the displayed equation (*) in the abstract, the right-hand side contains the typographical error 'div] v' instead of 'div v'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater precision in the proof of the main result. The observation is well-taken and we will revise the manuscript to include an explicit statement of the general interpolation theorem together with a verification that its hypotheses hold for the divergence operator.

read point-by-point responses

  1. Referee: [Proof of the main result (following the abstract)] The manuscript invokes a general theorem on interpolating solutions of linear equations to transfer the range equality (*). However, (*) supplies only existential (nonlinear) preimages under div, while real interpolation functors preserve bounded linear operators. Without an explicit linear selection or bounded right inverse for div on L^∞ ∩ X, it is unclear how the general theorem applies directly to produce solutions in L^∞ ∩ X_{θ,q}. This step is load-bearing for the claim and requires a precise statement of the general theorem together with verification that its hypotheses are met by the divergence operator under (*).

    Authors: We agree that the application of the general theorem requires careful justification and that the manuscript would be strengthened by making this step fully explicit. The theorem we rely on (a result on the interpolation of solvability for linear equations between Banach spaces) is formulated precisely to handle range equalities of the form div Y = div Z when Y and Z are compatible spaces; it does not require a linear right inverse, only the existence of solutions at the endpoint spaces together with the boundedness of the linear operator div. Nevertheless, we acknowledge that the current presentation leaves the precise hypotheses and their verification implicit. In the revised manuscript we will (i) state the general theorem verbatim, (ii) verify that the divergence operator satisfies all required hypotheses under the standing assumption (*), and (iii) explain how the interpolated solutions are obtained in L^∞ ∩ X_{θ,q}. This revision will be placed immediately after the statement of the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: claim follows from external general interpolation theorem applied to the given assumption (*).

full rationale

The derivation starts from the external Bourgain-Brezis result for W^{1,d} and the hypothesis that X satisfies the range equality (*). It then invokes a separate general theorem on interpolating solutions of linear equations to transfer (*) to the real interpolation spaces (L^∞, X)_{θ,q}. No step re-derives (*) from the interpolated spaces themselves, renames a fitted quantity, or reduces the conclusion to a self-citation chain whose base case is unverified. The argument is self-contained once the cited general method is granted; any doubt about whether that method requires a linear right inverse (absent from (*)) is a question of applicability, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the existence and applicability of a general interpolation method for solutions of linear equations such as the divergence equation; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption A general method exists that allows interpolation of solutions to linear equations between L^∞ and X.
    The proof is stated to be based on this method; it is invoked to transfer the (*) property to the interpolated spaces.

pith-pipeline@v0.9.0 · 5464 in / 1180 out tokens · 21967 ms · 2026-05-08T07:09:28.208441+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    An introduction, vol

    Bergh, J., L¨ ofstr¨ om, J.,Interpolation spaces. An introduction, vol. 223 in Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1976

    work page 1976

  2. [2]

    Bourgain, J.,Some consequences of Pisier’s approach to interpolation.Israel J. Math. 77, no. 1-2, 165–185, 1992

    work page 1992

  3. [3]

    Bourgain, J., Brezis, H.,On the equationdivX=fand application to control of phases. J.Amer. Math. Soc., 16(2) : 393-426 (electronic), 2003

    work page 2003

  4. [4]

    Bourgain, J., Brezis, H.,New estimates for elliptic equations and Hodge type systems. J. Eur. Math. Soc., 9, no. 2, 277-315, 2007

    work page 2007

  5. [5]

    Z., 1-2:, 427-460, 2013

    Bousquet, P., Mironescu, P., Russ, E.,A limiting case for the divergence equation.Math. Z., 1-2:, 427-460, 2013. 14

    work page 2013

  6. [6]

    Bousquet, P., Russ, E., Wang, Y., Yung, P-L.,Approximation in fractional Sobolev spaces and Hodge systems.J. Funct. Anal. 276, no. 5 (2019), 1430-1478

    work page 2019

  7. [7]

    and DeVore, R.,Harmonic analysis of the space BV

    Cohen, A., Dahmen, W., Daubechies, I. and DeVore, R.,Harmonic analysis of the space BV. Rev. Mat. Iberoamericana 19, no. 1, 235–263, 2003

    work page 2003

  8. [8]

    Curc˘ a, E.,On the interpolation of the spacesW l,1(Rd)andW r,∞(Rd), Rev. Mat. Iberoam. 40, no. 3, 931-986, 2024

    work page 2024

  9. [9]

    Curc˘ a, E.,Pathological sums of Sobolev spaces and some counterexamples, Math. Z. 311, 8, 2025

    work page 2025

  10. [10]

    Curc˘ a, E.,Some Bourgain-Brezis type solutions via complex interpolation, preprint 2024

    work page 2024

  11. [11]

    Instytut Matematyczny Polskiej Akademii Nauk

    Kislyakov, S.V., Kruglyak, N.Y.,Extremal problems in interpolation theory, Whitney- Besicovitch coverings, and singular integrals. Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) 74, Birkh ˜A¤user/Springer, Basel, 2013

    work page 2013

  12. [12]

    Kreyszig, E.,Introductory Functional Analysis with Applications, John Wiley & Sons, Inc., New York, 1978

    work page 1978

  13. [13]

    Banach J

    Lindemulder, N., Lorist, E.,Stein interpolation for the real interpolation method. Banach J. Math. Anal., 16:7, 2022

    work page 2022

  14. [14]

    Journal of Mathematical Physics

    Maligranda, L., Persson, L-E., Wyller, J.,Interpolation and partial differential equations. Journal of Mathematical Physics. 35. 5035-5046. 10.1063/1.530829., 1994

    work page doi:10.1063/1.530829 1994

  15. [15]

    Math., 445, 247-252, 2007

    Mazya, V.,Bourgain-Brezis type inequality with explicit constants, Contemp. Math., 445, 247-252, 2007

    work page 2007

  16. [16]

    Peetre, J.,Sur l’utilisation des suites inconditionellement sommables dans la th´ eorie des espaces d’interpolation. Rend. Sem. Mat. Univ. Padova 46, 173–190, 1971

    work page 1971

  17. [17]

    Michigan Math

    Pisier, G.,A simple proof of a theorem of Jean Bourgain. Michigan Math. J. 39 (1992), no. 3, 475–484

    work page 1992

  18. [18]

    Schmeisser, H.-J., Triebel, H.,Topics in Fourier analysis and function spaces, 300 pp., A Wiley-Interscience Publication, John Wiley & Sons Ltd., 1987

    work page 1987

  19. [19]

    Princeton University Press, Princeton, 2003

    Stein, E.M., Shakarchi, R.,Complex Analysis. Princeton University Press, Princeton, 2003. 15

    work page 2003