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arxiv: 1907.06380 · v1 · pith:ACLW5CILnew · submitted 2019-07-15 · 🧮 math.FA

Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces

Luigi D'Onofrio , Luigi Greco , Karl-Mikael Perfekt , Carlo Sbordone , Roberta Schiattarella This is my paper

Pith reviewed 2026-05-24 21:24 UTC · model grok-4.3

classification 🧮 math.FA
keywords atomic decompositiondual spaceBourgain-Brezis-Mironescu spaceBanach space dualitypredualsupremum normdistance formula
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The pith

A Banach space with norm equal to the supremum over a family of operators is the dual of a space whose predual admits an atomic decomposition; this applies directly to the Bourgain-Brezis-Mironescu space B.

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

The paper establishes that any Banach space E whose norm is defined exactly as the supremum of a given collection of operators must be a dual space, and that its predual admits an atomic decomposition. The argument is then specialized to the function space B of Bourgain, Brezis, and Mironescu, producing an atomic decomposition for the predual B_*, the identifications B_0^* = B_* and B_*^* = B, and an explicit formula for the distance from any element of B to the subspace B_0. These conclusions matter because they turn an abstract norm into concrete duality and approximation statements that can be used to compute distances and identify extremal elements without passing through the full dual ball.

Core claim

If the norm of a Banach space E is the supremum over a family of operators, then E is the dual of some Banach space whose unit ball is the closed convex hull of the atoms generated by those operators; the predual therefore carries an atomic decomposition. Applied to B this yields the biduality relations B_0^* = B_* and B_*^* = B together with the distance formula dist(f, B_0) expressed via the same family of operators.

What carries the argument

The supremum-type norm induced by a collection of operators, which supplies both the dual pairing and the atoms for the decomposition of the predual.

If this is right

  • B_0 is the predual of B_* and B is the dual of B_*
  • Every element of B admits an atomic decomposition coming from the predual B_*
  • The distance from f in B to B_0 equals the infimum of the operator-supremum expressions over the atoms
  • The same duality and decomposition statements hold for any other space whose norm is realized as a supremum over a fixed operator family

Where Pith is reading between the lines

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

  • The atomic decomposition may allow explicit computation of best approximations in B by finite sums of atoms
  • Similar supremum-norm constructions appear in other function spaces; the same argument would immediately give their duals and distance formulas
  • The biduality B_*^* = B suggests that B is weakly sequentially complete or has other structural properties that follow from being a dual

Load-bearing premise

The norm on E must be exactly equal to the supremum taken over the given family of operators.

What would settle it

An explicit Banach space whose norm equals the supremum over a concrete family of operators yet fails to be isometric to the dual of its predual, or whose predual admits no atomic decomposition with respect to those operators.

read the original abstract

Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\mathcal{B}_\ast$, the biduality result that $\mathcal{B}_0^\ast = \mathcal{B}_\ast$ and $\mathcal{B}_\ast^\ast = \mathcal{B}$, and a formula for the distance from an element $f \in \mathcal{B}$ to $\mathcal{B}_0$.

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 proves a general theorem: if a Banach space E has a norm that is the supremum induced by a given collection of operators, then E is a dual space and its predual admits an atomic decomposition. The authors apply this result together with prior work by one of the authors to the Bourgain-Brezis-Mironescu space B, obtaining an atomic decomposition of the predual B_*, the biduality statements B_0^* = B_* and B_*^* = B, and an explicit formula for the distance from f in B to the subspace B_0.

Significance. If the norm-identification step for B holds, the results supply concrete duality and atomic-decomposition tools for the BBM space and potentially for other spaces whose norms admit a similar supremum representation. The general theorem is a clean structural statement that could be reused; the distance formula is a concrete, falsifiable consequence.

major comments (1)
  1. [Application to B (post-general theorem)] Application section (likely §3 or §4): the transfer of the general duality and atomic-decomposition conclusions to B rests on an explicit verification that the BBM norm coincides exactly with the supremum over the stated operator family. The abstract asserts this identification, but the manuscript must supply the concrete operator collection and the density argument showing that the seminorm generated by the family equals the original B-norm on a dense subclass; without this step the conclusions do not follow from the general theorem.
minor comments (1)
  1. Notation for the predual and bidual spaces (B_*, B_0, B) should be introduced once with a clear diagram of the duality relations before the statements of the main theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the precise identification of the missing step in the application to the BBM space. We address the comment below and will revise the manuscript to make the argument fully self-contained.

read point-by-point responses

  1. Referee: Application section (likely §3 or §4): the transfer of the general duality and atomic-decomposition conclusions to B rests on an explicit verification that the BBM norm coincides exactly with the supremum over the stated operator family. The abstract asserts this identification, but the manuscript must supply the concrete operator collection and the density argument showing that the seminorm generated by the family equals the original B-norm on a dense subclass; without this step the conclusions do not follow from the general theorem.

    Authors: We agree that the explicit verification is required for the conclusions to follow directly from the general theorem. The identification of the BBM norm with a supremum over a suitable family of operators was established in prior work by one of the authors, but the current manuscript only cites that work without reproducing the concrete operator collection or the density argument. In the revised version we will add, in the application section, the explicit list of operators and a self-contained density argument (on a dense subclass such as C^∞ functions with compact support) showing that the induced seminorm coincides with the original B-norm. This will make the transfer of the duality and atomic-decomposition results fully rigorous within the paper. revision: yes

Circularity Check

0 steps flagged

General theorem for supremum-norm spaces plus prior author results applied to BBM space

full rationale

The paper states a general result: for a Banach space E whose norm is induced as the supremum over a given collection of operators, E is dual and its predual admits an atomic decomposition. This is then applied to the BBM space B together with 'some results obtained previously by one of the authors'. The central derivation (the general theorem) is self-contained and does not reduce to a fit, self-definition, or imported uniqueness theorem from the same authors. The self-citation supports only the auxiliary identification for B and is not load-bearing for the new general claim. No equation or step equates a derived quantity to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, axioms, or invented entities are visible in the provided text. The argument is described as resting on a supremum-norm representation and on earlier results by one of the authors.

pith-pipeline@v0.9.0 · 5683 in / 1243 out tokens · 20485 ms · 2026-05-24T21:24:55.064000+00:00 · methodology

discussion (0)

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Reference graph

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