arxiv: 2605.11295 · v1 · submitted 2026-05-11 · 🧮 math.FA

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Boundedness of the averaging projections in nonlocally convex Lorentz sequence spaces and applications to basis theory

Fernando Albiac, Jos\'e L. Ansorena, Miguel Berasategui

Pith reviewed 2026-05-13 00:46 UTC · model grok-4.3

classification 🧮 math.FA

keywords quasi-Banach spacesLorentz sequence spacessymmetric Schauder basesaveraging projectionslocal convexityconditional basesgreedy bases

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The pith

The canonical bases of weighted Lorentz sequence spaces have uniformly bounded averaging projections even without local convexity.

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

This paper shows that averaging projections tied to symmetric bases remain uniformly bounded in an entire class of nonlocally convex weighted Lorentz sequence spaces. These include the spaces ℓ_{p,q} for 0 < q < 1 < p < ∞, which are quasi-Banach but lack local convexity. A sympathetic reader would care because the property fails in other simple quasi-Banach spaces like ℓ_p for p < 1, suggesting it might mark local convexity, yet here it persists. The result implies that bounded averaging projections cannot characterize local convexity in quasi-Banach spaces with symmetric bases. It further delivers new instances of conditional bases and almost greedy bases in these nonconvex settings.

Core claim

Our main result shows that the canonical basis of an entire class of weighted Lorentz sequence spaces, including the spaces ℓ_{p,q} for 0<q<1<p<∞, has uniformly bounded averaging projections. Thus, bounded averaging projections do not characterize local convexity among quasi-Banach spaces with symmetric bases. As applications, we obtain new consequences for the structure of special bases. In particular, as a byproduct of our approach, we derive new examples of conditional and almost greedy bases in nonlocally convex spaces.

What carries the argument

Averaging projections associated with the symmetric Schauder basis in weighted Lorentz sequence spaces.

If this is right

The canonical basis remains a Schauder basis with the additional property of bounded averaging projections.
Local convexity is not necessary for this boundedness in quasi-Banach spaces with symmetric bases.
New examples of conditional bases exist in nonlocally convex Lorentz spaces.
New examples of almost greedy bases exist in nonlocally convex Lorentz spaces.

Where Pith is reading between the lines

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

This opens the possibility that other convexity-related properties might hold in these spaces without actual convexity.
Researchers could check if the same boundedness applies to bases that are not symmetric in these spaces.
One might investigate whether similar results hold for other nonlocally convex sequence spaces beyond Lorentz ones.

Load-bearing premise

The averaging projections considered are the standard ones associated with symmetric bases in the existing literature.

What would settle it

If the operator norm of these averaging projections grows without bound as one considers larger finite supports in the space ℓ_{p,q} for fixed p and q in the given range, that would disprove the main claim.

read the original abstract

We study the boundedness of averaging projections associated with symmetric Schauder bases in quasi-Banach spaces. Although this property is standard in the Banach setting, it is far from clear in the absence of local convexity and, indeed, fails for a broad class of quasi-Banach spaces with a symmetric basis, including $\ell_p$ for $0<p<1$. Our main result shows that, nevertheless, the canonical basis of an entire class of weighted Lorentz sequence spaces, including the spaces $\ell_{p,q}$ for $0<q<1<p<\infty$, has uniformly bounded averaging projections. Thus, bounded averaging projections do not characterize local convexity among quasi-Banach spaces with symmetric bases. As applications, we obtain new consequences for the structure of special bases. In particular, as a byproduct of our approach, we derive new examples of conditional and almost greedy bases in nonlocally convex spaces.

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.

Desk Editor's Note private letter to a colleague

The paper shows bounded averaging projections hold in weighted Lorentz spaces like ℓ_{p,q} (0<q<1<p<∞), countering the idea that they characterize local convexity in symmetric quasi-Banach spaces.

read the letter

The punchline is straightforward: bounded averaging projections associated with symmetric bases do not characterize local convexity among quasi-Banach spaces. The authors establish this by proving that the canonical basis in a broad family of weighted Lorentz sequence spaces has uniformly bounded averaging projections. This includes the spaces ℓ_{p,q} for parameters satisfying 0 < q < 1 < p < ∞, spaces that are known to be non-locally convex.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the canonical symmetric Schauder basis in weighted Lorentz sequence spaces, including the non-locally convex spaces ℓ_{p,q} for 0<q<1<p<∞, admits uniformly bounded averaging projections. This stands in contrast to the known failure of the same property for ℓ_p when 0<p<1. The authors conclude that bounded averaging projections therefore do not characterize local convexity among quasi-Banach spaces with symmetric bases, and they derive applications to the existence of conditional and almost greedy bases in these spaces.

Significance. If the quasi-norm estimates hold, the result supplies concrete counterexamples that separate bounded averaging projections from local convexity in the quasi-Banach setting. The explicit verification for an entire family of Lorentz spaces strengthens the literature on symmetric bases beyond the Banach case, and the byproduct constructions of conditional and almost greedy bases provide new examples useful for approximation theory in non-locally convex spaces.

minor comments (3)

The definition of the weighted Lorentz quasi-norm and the precise form of the averaging projections (standard symmetric averaging) should be recalled explicitly in §2 before the main estimates, to make the subsequent calculations self-contained for readers unfamiliar with the quasi-Banach literature.
In the applications to almost greedy bases, the dependence of the greedy constant on the averaging-projection bound should be stated more explicitly; a brief remark tracking how the uniform bound propagates would clarify the quantitative strength of the new examples.
A small number of typographical inconsistencies appear in the notation for the weight sequence (w_n) versus the Lorentz parameters (p,q); a uniform convention throughout the text would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. No specific major comments or criticisms were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; result follows from direct quasi-norm estimates

full rationale

The central claim is proved by explicit verification that the averaging projections are bounded on the weighted Lorentz spaces, using only the definition of the Lorentz quasi-norm, the symmetry of the canonical basis, and standard estimates for the associated operators. These steps are self-contained within the paper and do not reduce to fitted parameters, self-definitions, or load-bearing self-citations. The conclusion that bounded averaging projections fail to characterize local convexity is obtained simply by exhibiting the Lorentz spaces (including the non-locally convex ℓ_{p,q} family) as counterexamples. No derivation step equates a prediction to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definitions and properties of quasi-norms, symmetric Schauder bases, and rearrangement-invariant Lorentz sequence spaces that are already established in the functional-analysis literature; no new free parameters or invented entities appear in the abstract.

axioms (2)

standard math Quasi-Banach spaces satisfy a relaxed triangle inequality and are complete under the quasi-norm.
Invoked implicitly when discussing boundedness of operators on these spaces.
domain assumption Symmetric Schauder bases admit well-defined averaging projections.
Standard assumption in the theory of symmetric bases in sequence spaces.

pith-pipeline@v0.9.0 · 5471 in / 1480 out tokens · 82729 ms · 2026-05-13T00:46:15.679813+00:00 · methodology

discussion (0)

Reference graph

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