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arxiv: 1907.03439 · v1 · pith:LUSGNS2Enew · submitted 2019-07-08 · 🧮 math.AP

Sharp Logarithmic Sobolev and related inequalities with monomial weights

Filomena Feo , Futoshi Takahashi This is my paper

Pith reviewed 2026-05-25 01:20 UTC · model grok-4.3

classification 🧮 math.AP
keywords logarithmic Sobolev inequalitymonomial weightssharp constantsequality casesShannon inequalityHeisenberg uncertaintySobolev inequality
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The pith

A sharp logarithmic Sobolev inequality with monomial weights follows from the corresponding sharp Sobolev inequality.

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

The authors derive a sharp logarithmic Sobolev inequality that incorporates monomial weights directly from an assumed sharp Sobolev inequality carrying the same weights. The same starting point produces related sharp inequalities of Shannon type and Heisenberg uncertainty type. Equality cases receive an explicit characterization when every exponent in the monomial weights is either zero or a positive integer. The argument supplies a new proof of the result even in the classical unweighted setting.

Core claim

Starting from a sharp Sobolev inequality with monomial weights, the paper obtains the sharp logarithmic Sobolev inequality by a suitable limiting or scaling procedure. Equality cases are characterized when the monomial exponents are all zero or integers. The derivation simultaneously yields sharp Shannon-type and Heisenberg-type inequalities under the same weighted setting.

What carries the argument

Derivation of the logarithmic Sobolev inequality from the Sobolev inequality via a limiting procedure that preserves the monomial weights and their sharpness.

If this is right

  • Sharp Shannon-type entropy inequalities hold with the same monomial weights.
  • Heisenberg uncertainty principles hold sharply under the monomial weights.
  • Equality in the logarithmic Sobolev inequality is attained precisely for the functions identified when the exponents are zero or integers.
  • The same limiting argument recovers the classical unweighted logarithmic Sobolev inequality with its known equality cases.

Where Pith is reading between the lines

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

  • The method may apply to other entropy or uncertainty inequalities once a sharp Sobolev inequality is known in a given weighted space.
  • It offers a route to explicit constants in settings where direct variational methods are difficult.
  • The equality characterization could guide construction of extremals in related weighted functional inequalities.

Load-bearing premise

The derivation takes as given that a sharp Sobolev inequality with monomial weights already holds.

What would settle it

A test function for which the logarithmic Sobolev inequality is violated by the claimed sharp constant, while the underlying Sobolev inequality holds with equality.

read the original abstract

We derive a sharp Logarithmic Sobolev inequality with monomial weights starting from a sharp Sobolev inequality with monomial weights. Several related inequalities such as Shannon type and Heisenberg's uncertain type are also derived. A characterization of the equality case for the Logarithmic Sobolev inequality is given when the exponents of the monomial weights are all zero or integers. Such a proof is new even in the unweighted case.

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

2 major / 0 minor

Summary. The paper derives a sharp logarithmic Sobolev inequality with monomial weights from an assumed sharp Sobolev inequality with the same weights. It also obtains related inequalities of Shannon and Heisenberg uncertainty type. Equality cases in the logarithmic Sobolev inequality are characterized when all monomial exponents are zero or positive integers; the argument is asserted to be new even in the unweighted setting.

Significance. If the input Sobolev inequality is indeed sharp for general monomial weights, the derivation supplies a uniform method for obtaining sharp logarithmic Sobolev, Shannon, and uncertainty inequalities in the weighted setting, together with an explicit equality-case description for integer exponents. Such a reduction, if valid, would be a useful technical tool in weighted Sobolev theory.

major comments (2)
  1. [Abstract, §1] Abstract and §1: the claimed sharpness of the logarithmic Sobolev inequality is obtained directly by reduction from an assumed sharp Sobolev inequality with identical monomial weights; no independent proof or explicit reference establishing sharpness of the input inequality for arbitrary (non-integer) monomial exponents is supplied in the manuscript. Because the output sharpness inherits verbatim from the input, this assumption is load-bearing for the central claim.
  2. [Abstract] Abstract: the equality-case characterization is stated only for zero or integer exponents, yet the manuscript gives no indication whether the reduction itself produces additional equality cases or restrictions when the exponents are non-integer; this gap affects the completeness of the stated characterization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to improve clarity on the assumptions and scope of the results.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: the claimed sharpness of the logarithmic Sobolev inequality is obtained directly by reduction from an assumed sharp Sobolev inequality with identical monomial weights; no independent proof or explicit reference establishing sharpness of the input inequality for arbitrary (non-integer) monomial exponents is supplied in the manuscript. Because the output sharpness inherits verbatim from the input, this assumption is load-bearing for the central claim.

    Authors: The manuscript explicitly frames the logarithmic Sobolev inequality as derived from an assumed sharp Sobolev inequality with the same monomial weights, with the main contributions being the reduction method itself and the equality-case analysis for the integer/zero case. We agree that the sharpness claim is conditional on the input Sobolev inequality. We will revise the abstract and §1 to state more explicitly that all derived inequalities (including sharpness) hold under the hypothesis that the Sobolev inequality is sharp, and to note that establishing sharpness of the Sobolev inequality for general (non-integer) monomial weights lies outside the scope of this work. revision: partial

  2. Referee: [Abstract] Abstract: the equality-case characterization is stated only for zero or integer exponents, yet the manuscript gives no indication whether the reduction itself produces additional equality cases or restrictions when the exponents are non-integer; this gap affects the completeness of the stated characterization.

    Authors: The reduction establishes that equality holds in the logarithmic Sobolev inequality if and only if it holds in the underlying Sobolev inequality. The explicit characterization of equality cases is limited to zero and positive integer exponents because the argument identifying the extremals relies on properties (such as the form of monomials with integer powers) that are available only in those cases. For non-integer exponents the reduction does not generate additional equality cases or impose further restrictions beyond those inherited from the Sobolev inequality; since the latter are not characterized here, we restrict the statement accordingly. We will add a clarifying sentence in the abstract and §1 to this effect. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation from external sharp Sobolev input is non-circular

full rationale

The paper's central step is an explicit derivation of the sharp log-Sobolev inequality (and related forms) from a pre-existing sharp Sobolev inequality with the same monomial weights. This is a standard implication technique rather than a self-referential loop. No equations in the abstract or described structure reduce the output to a fit, self-definition, or renaming of the input. The equality-case characterization for zero/integer exponents is stated to be new even in the unweighted case, confirming independent content. If the input Sobolev inequality is established independently in the literature (as the abstract implies by taking it as given), the derivation chain remains self-contained and externally benchmarked. No load-bearing self-citation or ansatz smuggling is exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, ad-hoc axioms, or invented entities are identifiable. The work relies on standard background results in functional analysis.

axioms (2)
  • domain assumption Existence and sharpness of the Sobolev inequality with monomial weights (taken as starting point)
    The derivation explicitly starts from this inequality.
  • standard math Standard properties of Sobolev spaces and weighted measures in R^n
    Implicit in any such inequality paper.

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Reference graph

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