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arxiv: 1907.11327 · v1 · pith:FIVZGX3Lnew · submitted 2019-07-25 · 🧮 math.FA

Reverse Holder inequalities revisited: Interpolation, Extrapolation, Indices and Doubling

Alvaro Corval\'an , Mario Milman This is my paper

Pith reviewed 2026-05-24 15:40 UTC · model grok-4.3

classification 🧮 math.FA
keywords reverse Hölder inequalitiesweight classesK-functionalsSamko indexquasi-monotonicityextrapolation spacesL(p,q) normsnon-doubling measures
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The pith

Reverse Hölder weight classes are characterized by indices of K-functionals of the weights.

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

The paper characterizes the classical classes of weights satisfying reverse Hölder inequalities in terms of indices of suitable families of K-functionals of the weights. It introduces a Samko-type index for families of functions based on quasi-monotonicity and applies it to obtain index characterizations of the RH_p classes as well as the limiting class RH equal to the union over p greater than 1 of RH_p. This extends earlier results on interpolation and extrapolation and also covers inequalities with L(p,q) norms and non-doubling measures. A sympathetic reader would care because the indices tie the weight conditions directly to the structure of K-functionals.

Core claim

The classical classes of weights that satisfy reverse Hölder inequalities are characterized in terms of indices of suitable families of K-functionals of the weights. A Samko type of index for families of functions, based on quasi-monotonicity, is introduced and used to provide an index characterization of the RH_p classes, as well as the limiting class RH = RH_{LLogL} = ∪_{p>1} RH_p, which in the abstract case involves extrapolation spaces. Reverse Hölder inequalities associated to L(p,q) norms and non-doubling measures are also treated.

What carries the argument

Samko-type index for families of functions based on quasi-monotonicity, applied to K-functionals of the weights.

Load-bearing premise

The Samko-type index based on quasi-monotonicity can be defined for the relevant families of K-functionals without additional unstated restrictions on the weights or measures.

What would settle it

A concrete weight whose membership in some RH_p fails to match the value of the associated Samko-type index computed from its K-functionals.

read the original abstract

Extending results in \cite{M} and \cite{MM} we characterize the classical classes of weights that satisfy reverse H\"{o}lder inequalities in terms of indices of suitable families of $K-$functionals of the weights. In particular, we introduce a Samko type of index (cf. \cite{kara}) for families of functions, that is based on quasi-monotonicity, and use it to provide an index characterization of the $RH_{p}$ classes, as well as the limiting class $RH=$ $RH_{LLogL}=$. $\bigcup\limits_{p>1}RH_{p}$ (cf. \cite{BMR}),\ which in the abstract case involves extrapolation spaces. Reverse H\"{o}lder inequalities associated to $L(p,q)$ norms, and non-doubling measures are also treated.

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 extends results from [M] and [MM] by characterizing the classical reverse Hölder classes RH_p (p>1) and the limiting class RH = RH_{LLogL} = ∪_{p>1} RH_p in terms of indices of suitable families of K-functionals associated to weights. It introduces a Samko-type index for families of functions based on quasi-monotonicity, applies it to obtain index characterizations, treats associated inequalities for L(p,q) norms, and handles the abstract case via extrapolation spaces as well as the setting of non-doubling measures.

Significance. If the characterizations hold, the work supplies a coherent index-theoretic framework that unifies interpolation/extrapolation approaches to reverse Hölder inequalities, extending prior results to non-doubling measures and L(p,q) settings. The Samko-type index construction for families of K-functionals is a concrete technical contribution that may streamline proofs and comparisons across weighted-norm inequalities.

minor comments (4)
  1. [§2, Definition 2.3] §2, Definition 2.3: the quasi-monotonicity condition for the family {K(t,f)} is stated with a parameter δ>0; clarify whether the resulting index is independent of the choice of δ (or of the auxiliary function φ) under the standing assumptions on the weight class.
  2. [Theorem 4.2] Theorem 4.2 and Corollary 4.3: the statements equate the RH_p condition to an index inequality; add a brief remark on whether the constants implicit in the index are comparable to the RH_p constant, or whether the characterization is purely qualitative.
  3. [§5.1] §5.1, display (5.4): the extrapolation-space formulation for the limiting RH class contains an implicit supremum over p; make explicit that the index is taken with respect to the family indexed by p, to avoid ambiguity with the single-function Samko index.
  4. [References] References: [BMR] is cited for the limiting class but the precise statement used (e.g., the identification RH = RH_{LLogL}) is not restated; add a one-sentence recall of the relevant result from [BMR] for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. No specific major comments were listed in the report, so we have no points to address point-by-point. We are happy to incorporate any minor editorial suggestions if provided.

Circularity Check

0 steps flagged

Minor self-citation in extension of prior work; new index definition and characterizations are independent

full rationale

The paper explicitly extends results from [M] and [MM] but defines a new Samko-type index based on quasi-monotonicity for families of K-functionals and derives the RH_p and RH characterizations from this definition plus the interpolation/extrapolation framework. No load-bearing step reduces by construction to a fitted input, self-citation chain, or ansatz smuggled from prior work; the central claims rest on the newly introduced index and stated hypotheses on weights and measures. This is a standard case of building on one's own line of research without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5677 in / 1206 out tokens · 25351 ms · 2026-05-24T15:40:40.137884+00:00 · methodology

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