Trace type Orlicz spaces and analysis of Orlicz spaces by Lebesgue exponents

Albin Petersson

REVIEW 2 major objections 5 minor 13 references

Trace type Orlicz spaces and analysis of Orlicz spaces by Lebesgue exponents

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The Lebesgue exponents of a Young function are not invariants of its Orlicz space; the paper repairs this with equivalence-class exponents, logarithmic growth limits, and submultiplicativity.

desk verdict The exponent-moving construction for equivalent Young functions is new and correct; the Theorem 2.24 proof gap is real but repairable, and the paper deserves a proper referee. read the letter →

arxiv 2505.21764 v2 pith:VTRS6BBN submitted 2025-05-27 math.FA

Albin Petersson This is my paper

classification math.FA MSC 46E30

keywords OrliczspacesYoungfunctionsLebesgueexponentsDelta-2conditiontracetypemixedsubmultiplicativeinclusiontheorems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper concerns the two Lebesgue exponents $p_\Phi$ and $q_\Phi$ attached to a Young function $\Phi$, which bound the growth of $t\Phi'(t)/\Phi(t)$ and are often used to embed the Orlicz space $L^\Phi$ between Lebesgue spaces. The paper shows that these exponents are not a reliable signature of the space: whenever $1

What carries the argument

The machine that moves the argument is a piecewise power-splicing construction: replace parts of $\Phi$ by pure powers $ct^{r}$, glued with matching values and derivatives, to force the ratio $g_\Phi(t)=t\Phi'(t)/\Phi(t)$ to take prescribed values while keeping the new $\Psi$ equivalent to $\Phi$. The stabilization comes from the logarithmic ratio $r_\Phi(t)=\ln\Phi(t)/\ln t$, whose limits at $0$ and $\infty$ always exist under the $\Delta_2$ condition and are shared by all equivalent Young functions. In the trace-space part, the load-bearing identity is the comparison between the modular $\rho_\Phi(f)=\int\Phi(|f|)\,dx$ and $\Phi(\|f\|_{L^\Phi})$, which Proposition 3.8 ties to sub- and supermultiplicativity of $\Phi$.

What would settle it

Take a Young function $\Phi\in\Delta_2$ for which $r_0=\lim_{t\to0+}\ln\Phi(t)/\ln t$ and $r_\infty=\lim_{t\to\infty}\ln\Phi(t)/\ln t$ both exceed $1$ but $g_\Phi(t)=t\Phi'(t)/\Phi(t)$ oscillates without a limit, then test whether the inclusions $L^p\cap L^q\subseteq L^\Phi\subseteq L^p+L^q$ still hold for $p>\max\{r_0,r_\infty\}$ and $1<q<\min\{r_0,r_\infty\}$; a counterexample would disprove the theorem, and a proof that works despite the oscillation would confirm the conclusion while showing the construction needs repair.

Watch

Extended reading notes

Core claim

The central claim is that the ordinary Lebesgue exponents are unstable under replacing a Young function by an equivalent one, while two coarser objects are stable. Proposition 2.12 constructs, for any $\Phi$ with $1<p_1<q_\Phi\le p_\Phi<p_2<\infty$, an equivalent $\Psi$ with $q_\Psi\le p_1$ and $p_\Psi\ge p_2$, so the same Orlicz space can be generated by functions whose derivative-based growth exponents are almost arbitrary. The paper therefore defines class exponents $p_{[\Phi]}=\inf\{p_\Psi:\Psi\in[\Phi]\}$ and $q_{[\Phi]}=\sup\{q_\Psi:\Psi\in[\Phi]\}$ and shows that, when the limits of $t\Phi'(t)/\Phi(t)$ at $0$ and $\infty$ exist, these equal the maximum and minimum of the two limits (Corollary 2.19). It then proves that the logarithmic ratio $r_\Phi(t)=\ln\Phi(t)/\ln t$ always has limits $r_0,r_\infty$ for $\Phi\in\Delta_2$, and that these limits alone imply $L^p\cap L^q\subseteq L^\Phi\subseteq L^p+L^q$ for every $p>\max\{r_0,r_\infty\}$ and $1<q<\min\{r_0,r_\infty\}$ (Theorem 2.24). Finally, for trace Orlicz spaces, $L^{\Phi,\Phi}\subseteq L^\Phi$ holds if and only if $\rho_\Phi(f)\le C\Phi(\|f\|_{L^\Phi})$ for all $f$, equivalently if $\Phi$ is submultiplicative, with the reverse inclusion equivalent to the reversed inequality and equality equivalent to $L^\Phi=L^p$.

Load-bearing premise

The improved inclusion theorem assumes that the logarithmic growth limits near $0$ and $\infty$, which always exist, can be used in place of the derivative-based ratio $t\Phi'(t)/\Phi(t)$ in a construction that was only proved when that ratio has limits; if this substitution fails, Theorem 2.24 is left without a proof.

Editorial extensions

If this is right

For any $\Phi$ with finite derivative-based exponents, the pair $(q_\Phi,p_\Phi)$ cannot be used to certify that a space is $L^\Phi$; inclusion statements formulated with individual Lebesgue exponents must be checked against the equivalence-class versions.
For $\Phi\in\Delta_2$, the inclusions $L^p\cap L^q\subseteq L^\Phi\subseteq L^p+L^q$ are guaranteed for all $p>\max\{r_0,r_\infty\}$ and $1<q<\min\{r_0,r_\infty\}$, where $r_0,r_\infty$ are the limits of $\ln\Phi(t)/\ln t$; these limits always exist, so no separate regularity of $t\Phi'(t)/\Phi(t)$ is needed.
The trace space $L^{\Phi,\Phi}$ is a proper subset of $L^\Phi$ for submultiplicative non-power $\Phi$ such as $t+t^2$, and a proper superset for supermultiplicative $\Phi$ such as $t^3$ for $t<1$ and $t$ for $t\ge 1$.
$L^{\Phi,\Phi}=L^\Phi$ occurs exactly in the Lebesgue-space case, that is, when $\Phi$ is equivalent to $t^p$ for some $p\ge1$; consequently, for such $\Phi$, the modular inequalities $C^{-1}\Phi(\|f\|_{L^\Phi})\le\rho_\Phi(f)\le C\Phi(\|f\|_{L^\Phi})$ characterize the coincidence of the two spaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the logarithmic limits $r_0,r_\infty$ always exist for $\Phi\in\Delta_2$, a natural next step is to prove the inclusion theorem directly from those limits, avoiding any reliance on the derivative ratio $t\Phi'(t)/\Phi(t)$; this would make the improvement fully self-contained.
The instability result invites the same equivalence-class check for other growth indices used in Orlicz theory, since any index that changes within $[\Phi]$ cannot be a reliable invariant of $L^\Phi$.
The trace-space characterization offers a practical test for whether a mixed-norm modulation space coincides with its full Orlicz counterpart: examine the submultiplicativity of the relevant Young function, since the paper shows equality fails in general.
A natural class-level version of submultiplicativity would ask whether some representative of $[\Phi]$ is submultiplicative; this would connect the trace-space inclusion to equivalence classes rather than to a single representative.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The genuinely new material is Section 2, and it holds up. Proposition 2.12—the claim that within an equivalence class of Young functions you can push q_Ψ below any p1 > 1 and p_Ψ above any p2 < ∞ while keeping L^Ψ = L^Φ—is the advertised result, and the splice construction works. I checked the constants the way you did: k < 0 from r1 < q_Φ ≤ g_Φ(1), l_α > 0 from r2 > p_Φ ≥ g_Φ(β), and the tail bounds give the exponent estimates. The pointwise equivalence to Φ is fine because the spliced intervals are finite and Φ is positive. So the central claim is solid, and it is a genuine clarification: p_Φ and q_Φ are not equivalence invariants, and the paper quantifies exactly how much they can move.

The equivalence-class exponents p_[Φ], q_[Φ] in Section 2.2 are a natural response, and the paper is right that they give sharper inclusions than (1.3) when the limits of g_Φ exist (Prop 2.20). The shift to r_Φ(t) = ln Φ(t)/ln t in Section 2.3 is sensible, and Prop 2.22's claim that r0 and r∞ always exist for Φ ∈ Δ₂ is a real simplification. The reliance on [3] for background is fine; the new constructions are self-contained.

The soft spots are real but not fatal. Theorem 2.24's proof invokes the Proposition 2.18 construction, and Proposition 2.18 needs the limits of g_Φ to exist. Theorem 2.24 only supplies limits of r_Φ, and those are different objects—r_Φ can converge while g_Φ oscillates. That is a genuine gap in the proof as written. The stress-test note has the right fix: the power bounds in Prop 2.22(iv) deliver the same inclusion without g_Φ limits, so this is a repairable presentation flaw, not a threat to the main result. Minor point: Prop 2.12 assumes Φ(1) > 0 and Φ'(1) > 0 without comment. Both are automatic for Φ ∈ Δ₂, but the paper should say so.

Section 3 is mostly consolidation. The equivalences between trace-type inclusions and sub/supermultiplicativity are credited to Maligranda [9] and Finol-Maligranda [4], and the self-contained proofs are fine. Example 3.1's computation of ||f_n||_{L2,1} is garbled in the text and the displayed equality is not justified as printed; the example is illustrative and the claim is plausible, but a referee should insist on a clean computation.

Overall, this is for people who work with Orlicz space indices and inclusions: a useful new construction plus a clean framework for class-level exponents. It deserves a serious referee. I would send it out, expecting a revise-and-resubmit with the Theorem 2.24 proof fixed and Example 3.1 cleaned up.

Referee Report

2 major / 5 minor

Summary. The paper analyzes the Lebesgue exponents pΦ and qΦ of Young functions and studies what information they carry about the equivalence class of the Young function and about inclusions between Orlicz spaces. The central construction (Proposition 2.12) shows that, under 1<p1<qΦ≤pΦ<p2<∞, one can find an equivalent Young function Ψ with qΨ≤p1 and pΨ≥p2, so the exponents are not invariant under equivalence. This motivates class-level exponents (Section 2.2) and an rΦ(t)=lnΦ(t)/ln t approach (Section 2.3) leading to improved inclusion statements L^p∩L^q⊆L^Φ⊆L^p+L^q. Section 3 characterizes, for Φ∈Δ2, inclusions between trace-type mixed Orlicz spaces L^{Φ,Φ} and L^Φ in terms of submultiplicativity, supermultiplicativity, and inequalities between ρΦ(f) and Φ(∥f∥_{L^Φ}).

Significance. If the results are correct, the paper gives a clean, explicit answer to a natural question: the common Lebesgue exponents qΦ and pΦ of a Young function are far from determined by the Orlicz space L^Φ, and can be moved almost arbitrarily within an equivalence class by an elementary splice construction. The class-level exponents and the rΦ-based inclusions are useful refinements, and the trace-type characterizations in Section 3 are elegant and connect to known results such as [4] and [9]. The paper is largely self-contained and provides concrete examples and explicit constants in the main construction, which are genuine strengths. The main advertised exponent-moving claim (Proposition 2.12) appears sound; the proof of Theorem 2.24 and one step in the proof of Proposition 3.12, however, contain gaps that need repair.

major comments (2)

[Section 2.3, Theorem 2.24 proof] The proof states that by 'the exact same construction as in the proof of Proposition 2.18 (now with r∈[q,p])' one obtains an equivalent Young function Ψε with p≤pΨε<p+ε and q−ε<qΨε≤q. Proposition 2.18 is proved under the hypothesis that the limits of gΦ(t)=tΦ′(t)/Φ(t) at 0 and at infinity exist. Theorem 2.24, however, supplies only limits of rΦ(t)=lnΦ(t)/ln t. These are different quantities: by Proposition 2.22(ii), r0 and r∞ need only lie between the liminf and limsup of gΦ, so gΦ may oscillate while rΦ converges. Thus the existence of Ψε is not established by the cited argument. The theorem itself is likely correct, since Proposition 2.22(iv) gives power bounds t^{r∞−ε}<Φ(t)<t^{r∞+ε} for large t and analogous bounds near zero from which the inclusions can be derived directly, but the proof as written is incomplete.
[Section 3.2, Proposition 3.12(i) proof] In the converse direction of the proof of Proposition 3.12(i), condition (C1) is Φ(C1ab)≤Φ1(a)Φ2(b). The proof sets a=|f(x,y)|/∥f(·,y)∥_{L^{Φ1}} and b=∥f(·,y)∥_{L^{Φ1}} and then writes Φ(|f(x,y)|)≤C1Φ1(...)Φ2(...). This does not follow by substitution; substitution gives Φ(C1|f(x,y)|)≤Φ1(...)Φ2(...). The gap is repairable: one obtains ρΦ(C1f)≤CρΦ2(∥f(·,y)∥_{L^{Φ1}}), and since L^Φ is a vector space, finiteness of ρΦ(C1f) implies f∈L^Φ. The statement itself is cited to [9], but the self-contained proof as written is invalid and should be corrected.

minor comments (5)

[Definition 1.7] There is a typo: 'are said the be equivalent' should read 'are said to be equivalent'.
[Section 2.2, proof of Proposition 2.18] The symbols p1 and p2 are used in the proof but have not been defined; from context they appear to mean p0=lim_{t→0}gΦ(t) and p∞=lim_{t→∞}gΦ(t). Please define them explicitly and align the notation with the statement.
[Example 3.2] The step 'Since LΦ,Φ⊊LΦ, it therefore follows by duality that LΨ⊊LΨ,Ψ' needs justification. Strict inclusion alone does not automatically imply strict inclusion of the duals; one needs to argue about density and the injectivity of the restriction map.
[Example 3.1] The Gaussian integral computations for ∥fn∥_{L^{2,1}} are very terse; adding the variable substitutions and intermediate steps would make the example reproducible and easier to verify.
[Section 2.3, after definition of aΦ,bΦ] The phrase 'which is (possibly) an improvement of Corollary 2.6(iii)' would be clearer if it specified in what sense aΦ,bΦ can be sharper than qΦ,pΦ, since both statements concern two-sided power bounds.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the exponent-moving construction is self-contained, with only a minor non-load-bearing self-citation in the background.

full rationale

The central advertised claim, Proposition 2.12, is proved by an explicit splice construction rather than by fitting or renaming inputs. The constants c_i, d_i are solved from continuity of Psi and Psi' at the breakpoints t=1, alpha, beta, and the bounds on q_Psi and p_Psi are read directly from the piecewise formula for t Psi'(t)/Psi(t). The conclusion that q_Psi < p1 and p_Psi > p2 follows from taking alpha and beta large, not from assuming the conclusion. Proposition 2.18 and Corollary 2.19 are likewise constructive: given limits of g_Phi(t) at zero and infinity, the paper builds equivalent Young functions Psi with p_Psi and q_Psi close to those limits, and then defines the class exponents p[Phi] and q[Phi] as infimum and supremum over that construction. No fitted parameter is relabelled as a prediction. The Section 3 equivalences are proved in both directions using indicator test functions and modular inequalities (Propositions 3.8 and 3.12), with only standard external citations for known parts; the main equivalence claims do not reduce to the paper's own prior work. The only self-citation is Proposition 2.15, which cites [3, Corollary 2.8] from the author's joint prior paper. That citation is not load-bearing: the invariance of p_Phi < infinity under pointwise equivalence already follows from Remark 2.4 within the present paper, and Proposition 2.15 is not used to force the main constructions. The reader-flagged gap in Theorem 2.24 is a genuine proof overstatement, since the construction in Proposition 2.18 assumes limits of g_Phi(t), whereas Theorem 2.24 assumes only limits of r_Phi(t); however, this is a repairable presentation issue rather than a circularity, and it does not affect the main exponent-moving construction. Score 2 reflects the minor non-load-bearing self-citation, not any reduction of a derivation to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new physical or mathematical objects are postulated. Definition 2.14 introduces class-level exponents, but these are derived from the existing equivalence relation, not independent entities. No empirical or data-fitted parameters appear in the paper.

assumptions (4)

standard math Standard Orlicz space theory: Luxemburg norm, complementary Young functions, Delta2 condition, duality of Orlicz spaces.
Used throughout for definitions, norm-modular equivalences in Propositions 3.3 and 3.7, and duality in Example 3.2.
standard math Characterization of inclusions between L^Phi and L^{Phi1,Phi2} in Proposition 3.12 from Maligranda [9], and the equality case from Finol-Maligranda [4].
The paper cites these sources; it does not derive them independently.
domain assumption The assumption Phi in Delta2 for all results in Section 3 and for the limit existence in Proposition 2.22.
Many statements are stated only for Phi in Delta2, equivalently p_Phi < infinity.
standard math The monotonicity argument in Proposition 2.22 that the threshold r separating limits 0 and infinity of Phi(t)/t^alpha is well-defined.
This is a standard property of the Karamata-type order of growth, used to prove existence of the limits of r_Phi.

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Cite this review

Pith. "Pith review of Trace type Orlicz spaces and analysis of Orlicz spaces by Lebesgue exponents." pith.science (2026). https://pith.science/paper/VTRS6BBN

@misc{pith2026250521764,
  author       = {Pith},
  title        = {Pith review of: Trace type Orlicz spaces and analysis of Orlicz spaces by Lebesgue exponents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTRS6BBN}},
  note         = {Machine review of arXiv:2505.21764}
}

abstract

In the paper, we analyze the Lebesgue exponents $p_\Phi$ and $q_\Phi$, and show that for any $p_\Phi< p < \infty$ and $1< q<q_\Phi$, there exists an equivalent Young function $\Psi$ with $p < p_\Psi < \infty$ and $1<q_\Psi < q$. This type of construction is used to improve upon the inclusions $L^{p_\Phi}\cap L^{q_\Phi}\subseteq L^\Phi \subseteq L^{p_\Phi} + L^{q_\Phi}$. For trace type Orlicz spaces $L^{\Phi,\Phi}$, we find that when $\Phi \in \Delta_2$, we have $L^{\Phi,\Phi} \subseteq L^\Phi$ if and only if $\Phi(||f||_{L^\Phi}) \le C \rho_\Phi(f)$ for all $f\in L^\Phi$, and the reverse inclusion is equivalent to the reversed inequality.

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Works this paper leans on

13 extracted references · 13 canonical work pages

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M. Bonino, S. Coriasco, A. Petersson, J. Toft, Fourier Type Operators on Orlicz Spaces and the Role of Orlicz Lebesgue Exponents,Mediterr. J. Math.21, (2024), art. no. 219

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [3]

Bonino, S

M. Bonino, S. Coriasco, A. Petersson, J. Toft, Fourier Type Operators on Orlicz Spaces and the Role of Orlicz Lebesgue Exponents,Mediterr. J. Math.21, (2024), art. no. 219

work page 2024

[2] [9]

Maligranda, Calderón–Lozanovskii; construction for mixed norm spaces, Acta Math

L. Maligranda, Calderón–Lozanovskii; construction for mixed norm spaces, Acta Math. Hung.103(4), (2004), 279–302

work page 2004

[3] [4]

Finol, L

C. Finol, L. Maligranda, On a decomposition of some functions,Comment. Math. Prace Mat.30(2), (1991), 285–291

work page 1991

[4] [1]

R. A. Adams, J. J. F. Fournier,Sobolev spaces, 2nd ed., Academic Press, 2008

work page 2008

[5] [2]

Birnbaum, W

Z. Birnbaum, W. Orlicz, Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen (German),Stud. Math.3(1), (1931), 1–67

work page 1931

[6] [5]

W. A. J. Luxemburg, Banach Function Spaces, PhD thesis, Technische Hogeschool te Delft, 1955

work page 1955

[7] [6]

W. A. Majewski, L. E. Labuschagne, On Applications of Orlicz Spaces to Statistical Physics, Ann. Henri Poincaré15(6), (2013), 1197–1221

work page 2013

[8] [7]

Maligranda, Indices and interpolation,Diss

L. Maligranda, Indices and interpolation,Diss. Math.234, (1985), 1–54

work page 1985