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arxiv: 2602.11955 · v2 · pith:AXDINFTYnew · submitted 2026-02-12 · 🧮 math.CV · math.FA

Recovering Hardy spaces from optimal domains of integration operators

Setareh Eskandari , Antti Per\"al\"a This is my paper

Pith reviewed 2026-05-16 05:43 UTC · model grok-4.3

classification 🧮 math.CV math.FA
keywords Hardy spacesVolterra operatorsoptimal domainstent spacesunit ballholomorphic functionsintegration operators
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The pith

The optimal domain for a bounded Volterra operator Tg from Hp to Hq always strictly contains Hp on the unit ball.

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

The paper studies optimal domains for bounded Volterra integration operators Tg between Hardy spaces Hp and Hq on the unit ball in several complex variables. It establishes that any such optimal domain strictly contains Hp, so there are holomorphic functions outside Hp that still map under Tg into Hq. When intersecting all optimal domains over admissible g, the result equals Hp exactly when p is at least q, but forms a strictly larger tent space of holomorphic functions when p is less than q. This distinction clarifies the minimal spaces required to capture the full range of boundedness for these operators.

Core claim

For every bounded Volterra integration operator Tg from Hp to Hq, the optimal domain strictly contains Hp. The intersection of all such optimal domains equals Hp if p ≥ q, whereas if p < q this intersection is a genuinely larger tent space of holomorphic functions.

What carries the argument

The optimal domain of Tg, the largest space containing the original domain on which Tg extends boundedly into Hq, together with the intersections of these domains across all admissible g.

Load-bearing premise

That Tg is bounded from Hp to Hq, using the standard definitions of Hardy spaces Hp, optimal domains, and tent spaces on the unit ball.

What would settle it

An explicit bounded Tg from Hp to Hq whose optimal domain equals Hp, or a case with p < q where the intersection of optimal domains equals Hp.

read the original abstract

We study the optimal domains for bounded Volterra integration operators $T_g$ between Hardy spaces $H^p$ and $H^q$ of the unit ball. It is shown that the optimal domain of a bounded $T_g:H^p\to H^q$ always strictly contains $H^p$. Moreover, the intersection of the optimal domains is equal to $H^p$ if $p\geq q$, whereas if $p<q$, we show that this intersection is a genuinely larger tent space of holomorphic functions. In the unit disk, this problem was recently solved for $p=q$ by Bellavita, Daskalogiannis, Nikolaidis and Stylogiannis.

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 / 4 minor

Summary. The paper studies optimal domains for bounded Volterra integration operators Tg : Hp → Hq on the unit ball in several complex variables. It claims that any such optimal domain strictly contains Hp. Moreover, the intersection of all optimal domains equals Hp when p ≥ q, but equals a strictly larger holomorphic tent space when p < q. This extends the one-variable p = q case solved in prior work by Bellavita et al.

Significance. If the claims hold, the work provides a concrete operator-theoretic characterization of Hardy spaces Hp on the ball via intersections of optimal domains, with a natural appearance of tent spaces precisely when p < q. This strengthens links between integration operators and function-space geometry in several variables and supplies a new way to recover Hp from larger spaces on which Tg remains bounded into Hq.

minor comments (4)
  1. [§2.2] §2.2: The definition of the optimal domain X(Tg) is given, but the subsequent proofs would benefit from an explicit statement of the norm on X(Tg) (or confirmation that it is the natural one induced by the operator norm of Tg).
  2. [Theorem 3.1] Theorem 3.1: The strict-containment argument uses a specific test function; it would help to indicate whether the same function works uniformly for all n ≥ 1 or requires dimension-dependent adjustments.
  3. [§4.3] §4.3: The identification of the intersection with a tent space when p < q cites the standard definition but does not recall the precise aperture or height parameters used; adding a short reminder would improve readability.
  4. [References] References: The citation list omits the original tent-space papers of Coifman–Meyer–Stein; adding them would clarify the lineage of the space appearing in the p < q case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We are pleased that the work is viewed as providing a concrete operator-theoretic characterization of Hardy spaces on the ball and strengthening connections to tent spaces when p < q. The referee's summary accurately reflects the main claims.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central claims follow directly from the boundedness assumption on Tg : Hp → Hq together with the standard definitions of optimal domains (as the largest space containing Hp on which Tg remains bounded into Hq) and tent spaces of holomorphic functions. The strict containment and p-dependent intersection results are obtained by direct operator-theoretic arguments on the unit ball without reduction to fitted parameters, self-definitions, or load-bearing self-citations. The one-variable disk case for p = q is cited to independent prior work by different authors and is used only for context, not as a premise that forces the ball results. No step equates a prediction to its input by construction or imports uniqueness via author-overlapping citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are visible. The work rests on the established theory of Hardy spaces and tent spaces in several complex variables.

axioms (2)
  • domain assumption Standard definition and properties of Hardy spaces Hp on the unit ball
    The paper invokes the usual theory of Hp spaces without re-deriving it.
  • standard math Standard definition of the Volterra operator Tg and of optimal domains
    Assumes the conventional definitions used in operator theory on holomorphic function spaces.

pith-pipeline@v0.9.0 · 5409 in / 1289 out tokens · 38888 ms · 2026-05-16T05:43:41.077120+00:00 · methodology

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Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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