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arxiv: 2605.17663 · v1 · pith:JPJKUDBJnew · submitted 2026-05-17 · 🧮 math.CA · math.FA

A Banach space that distinguishes two maximal operators

Vjekoslav Kova\v{c} This is my paper

Pith reviewed 2026-05-19 22:02 UTC · model grok-4.3

classification 🧮 math.CA math.FA
keywords maximal operatorsBanach spacestranslation-invariantM^diamondM^sharpMaz'ya-Shaposhnikovaopen problems
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The pith

There exists a translation-invariant Banach space of locally integrable functions on which M^diamond is bounded but the sharp maximal operator M^sharp is not.

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

The paper constructs a translation-invariant Banach space of locally integrable functions to show that the non-classical maximal operator M^diamond is bounded on it while the sharp maximal operator M^sharp is not. This construction directly addresses and answers one of Maz'ya's open questions from a list of 75 problems in analysis. A sympathetic reader cares because it demonstrates that these two operators are fundamentally different in their boundedness properties on Banach spaces, which has implications for how maximal operators are used in real analysis. The example is concrete and can be studied further to understand the differences between classical and non-classical maximal operators.

Core claim

We construct a translation-invariant Banach space of locally integrable functions on which M^diamond is bounded, but the sharp maximal operator M^sharp is not. This answers one of Maz'ya's questions from a collection of 75 open problems in analysis.

What carries the argument

The translation-invariant Banach space of locally integrable functions on which the boundedness of M^diamond holds but that of M^sharp does not.

If this is right

  • Answers Maz'ya's open problem on distinguishing the operators.
  • Shows that M^diamond and M^sharp are not equivalent for boundedness on all such spaces.
  • Provides a concrete example for further investigation in harmonic analysis.

Where Pith is reading between the lines

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

  • This space might have additional properties worth studying, such as its dual or embedding into other spaces.
  • Similar constructions could be attempted for other pairs of operators to see if distinctions are common.

Load-bearing premise

The space is a genuine Banach space that is translation-invariant and the boundedness and unboundedness claims for the two operators are valid.

What would settle it

Direct check that the defined space satisfies the Banach space axioms, is translation-invariant, and that the operator norm of M^diamond is finite while that of M^sharp is infinite.

read the original abstract

Maz'ya and Shaposhnikova introduced a non-classical maximal operator $M^\diamond$ as the maximal convolution with the vector-valued signum kernel truncated to centered balls. We construct a translation-invariant Banach space of locally integrable functions on which $M^\diamond$ is bounded, but the sharp maximal operator $M^\sharp$ is not. This answers one of Maz'ya's questions from a collection of 75 open problems in analysis.

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 / 2 minor

Summary. The manuscript constructs a translation-invariant Banach space of locally integrable functions on which the non-classical maximal operator M^diamond (defined via maximal convolution with a vector-valued signum kernel truncated to centered balls, following Maz'ya-Shaposhnikova) is bounded, while the sharp maximal operator M^sharp is unbounded. This explicit construction answers one of Maz'ya's open questions from a list of 75 problems in analysis.

Significance. If the construction holds, the result is significant because it furnishes a concrete, translation-invariant Banach space separating the boundedness properties of M^diamond and M^sharp. The paper receives credit for delivering an explicit construction rather than an abstract existence argument, directly resolving a specific open problem in the theory of maximal operators.

minor comments (2)
  1. §1 (Introduction): recall the precise definitions of M^diamond and M^sharp with direct citations to Maz'ya-Shaposhnikova to ensure the distinction is self-contained for readers unfamiliar with the 75-problem list.
  2. The norm estimates in the verification that the constructed space is complete and translation-invariant could be collected in a single lemma for easier reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, for recognizing the significance of our explicit construction, and for recommending minor revision. The report accurately captures that we furnish a concrete translation-invariant Banach space separating the boundedness properties of M^♦ and M^♯, directly resolving one of Maz'ya's open questions. No specific major comments appear in the report.

Circularity Check

0 steps flagged

Explicit construction with no circularity in derivation chain

full rationale

The paper presents an explicit construction of a translation-invariant Banach space of locally integrable functions on which M^diamond is bounded but M^sharp is not, directly answering an open question of Maz'ya. This is an existence result via concrete construction rather than any derivation, prediction, or ansatz that reduces to its own inputs by definition or self-citation. No load-bearing steps of the enumerated circular kinds appear; the boundedness claims rest on verifying the space's properties under the given operator definitions, which are external to the construction itself. The result is self-contained against external benchmarks and does not rely on fitted parameters renamed as predictions or uniqueness theorems imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests primarily on the existence and properties of the newly constructed space together with the standard definitions of the two maximal operators from prior work.

axioms (1)
  • standard math Standard axioms of Banach spaces, translation invariance, and local integrability of functions.
    These are background facts from functional analysis invoked to define the space and the operators.
invented entities (1)
  • The specific translation-invariant Banach space no independent evidence
    purpose: To serve as a counterexample separating boundedness of M^diamond and M^sharp
    The space is defined within the paper for this purpose; no independent existence proof outside the construction is given in the abstract.

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Reference graph

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