{"id":"e9a4b538-171c-435e-ac5f-0236e425a612","arxiv_id":"2505.21764","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs equivalent Young functions with arbitrary Lebesgue exponents and characterizes when trace-type Orlicz spaces equal ordinary Orlicz spaces.","lead":"This paper studies Orlicz spaces, a flexible family of function spaces generalizing the classical L^p spaces. It shows the standard growth exponents of the defining function can be altered without changing the space, and it characterizes when a mixed-norm version equals the original.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection to the central exponent-moving claim (Prop. 2.12); the reader's Theorem 2.24 proof gap is real but repairable and does not affect the main construction.","rationale":"The reader's weakest assumption is about Theorem 2.24, not about the strongest claim. I agree that the proof of Theorem 2.24 contains a real gap: Proposition 2.18 requires the limits of gΦ(t)=tΦ′(t)/Φ(t) at 0 and ∞, while Theorem 2.24 only assumes the limits of rΦ(t)=lnΦ(t)/ln t. These quantities can behave differently, so the reduction as written is unjustified. However, this gap does not threaten Proposition 2.12, which is the central exponent-moving claim and which I find internally consistent. Moreover, Theorem 2.24 itself appears true: the power bounds in Proposition 2.22(iv) directly give Lp∩Lq⊆LΦ and LΦ⊆Lp+Lq for the stated p and q by splitting f into large-value and small-value parts. Thus the reader's CONDITIONAL verdict is appropriate but the concern is a proof repair, not a correctness failure of the main construction. Separately, Example 3.1 contains an equality ∥∑n^{-5/4}f_n∥_{L2,1}=Σn^{-5/4}∥f_n∥_{L2,1} that is not justified and may be false; this is a secondary issue in the trace-type section and does not affect the central claim about Lebesgue exponents.","tokens_in":23416,"tokens_out":35980,"duration_ms":380587,"concrete_test":"Verify Proposition 2.12 numerically for Φ(t)=t^2, p1=1.5, p2=3: choose r1=1.2, r2=3.5, solve for α from h(α)=p1 and for β from pΨ=p2, then check (a) Ψ(t)/t^2 is bounded away from 0 and infinity, (b) qΨ≤1.5, and (c) pΨ≥3. Also re-derive Theorem 2.24 directly from Proposition 2.22(iv); if the inclusion follows, the flawed reduction to Proposition 2.18 is non-load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised claim that Lebesgue exponents can be moved within an equivalence class is Proposition 2.12. I checked the splice construction line by line: the constants in the proof make Ψ and Ψ′ continuous at t=1, α, β; k<0 follows from r1<qΦ≤gΦ(1); lα>0 follows from r2>pΦ≥gΦ(β) and r1<r2; the claimed bounds qΨ≤r1/(1+kα^{−r1})→r1<p1 and pΨ≥r2/(1+lαβ^{−r2})→r2>p2 are valid. Pointwise equivalence to Φ holds because Φ is positive, the spliced intervals are finite, and Ψ(t)/Φ(t) is bounded on the unbounded tails. So I do not find a load-bearing defect in the central claim. The proof of Theorem 2.24 does overstate when it invokes the Proposition 2.18 construction under only the rΦ-limits: gΦ(t)=tΦ′(t)/Φ(t) need not converge even if lnΦ(t)/ln t does. This is a genuine gap in that proof, but Theorem 2.24 can be proved directly from the power bounds in Proposition 2.22(iv), so it is a repairable presentation issue rather than a threat to the main exponent-moving result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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}).","tokens_in":23642,"tokens_out":13698,"duration_ms":140056,"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":[{"comment":"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":"Section 2.3, Theorem 2.24 proof"},{"comment":"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.","section":"Section 3.2, Proposition 3.12(i) proof"}],"minor_comments":[{"comment":"There is a typo: 'are said the be equivalent' should read 'are said to be equivalent'.","section":"Definition 1.7"},{"comment":"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.","section":"Section 2.2, proof of Proposition 2.18"},{"comment":"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.","section":"Example 3.2"},{"comment":"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":"Example 3.1"},{"comment":"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.","section":"Section 2.3, after definition of aΦ,bΦ"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nSection 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.\n\nOverall, 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.","headline":"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.","tokens_in":24188,"tokens_out":9472,"would_cite":true,"duration_ms":76294,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Orlicz spaces","Young functions","Lebesgue exponents","Delta-2 condition","trace type Orlicz spaces","mixed Orlicz spaces","submultiplicative functions","inclusion theorems"],"falsifier":"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.","tokens_in":23165,"feed_emoji":"📐","tokens_out":11920,"duration_ms":105906,"temperature":0.7,"pith_summary":"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 $1<p_1<q_\\Phi\\le p_\\Phi<p_2<\\infty$, one can construct an equivalent Young function $\\Psi$ with $L^\\Psi=L^\\Phi$ whose exponents fall outside the interval $[p_1,p_2]$. To repair the damage, it introduces exponents defined on the whole equivalence class $[\\Phi]$ and proves that the logarithmic ratios $\\ln\\Phi(t)/\\ln t$ have limits at $0$ and $\\infty$ for every $\\Phi$ with $p_\\Phi<\\infty$, yielding stable inclusions $L^p\\cap L^q\\subseteq L^\\Phi\\subseteq L^p+L^q$. In the second part, for $\\Phi\\in\\Delta_2$, the trace-type mixed space $L^{\\Phi,\\Phi}$ (the same Young function applied separately to two groups of variables) is shown to lie inside $L^\\Phi$ exactly when $\\Phi$ is submultiplicative, to contain it exactly when $\\Phi$ is supermultiplicative, and to coincide with it only when $L^\\Phi$ is a Lebesgue space $L^p$.","feed_headline":"Same Orlicz space admits wildly different growth exponents","feed_subtitle":"Logarithmic growth limits restore stable Lebesgue-space inclusions and pin down when trace Orlicz spaces agree.","key_machinery":"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 $\\Phi$.","core_discovery":"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 $L^\\Phi=L^p$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Lebesgue exponents and supplies the comparison inequalities and equivalence results that Section 2 refines.","marker":"[3]"},{"why":"Provides the index bounds and limit-invariance arguments used in Proposition 2.1, Lemma 2.17, and related steps.","marker":"[7]"},{"why":"Gives the mixed-norm inclusion criteria on which Proposition 3.12 and the trace-space characterizations rest.","marker":"[9]"},{"why":"Establishes that equality of $L^{\\Phi,\\Phi}$ and $L^\\Phi$ occurs only in the Lebesgue-space case.","marker":"[4]"},{"why":"Supplies the submultiplicativity and supermultiplicativity framework and the classification of Young functions that are both.","marker":"[11]"},{"why":"Introduces the Lebesgue exponents whose instability is the paper's point of departure.","marker":"[12]"}],"fun_headline_variants":["Orlicz exponents twist under equivalent Young functions","Trace Orlicz inclusion boils down to submultiplicativity","Logarithmic ratio stabilizes Orlicz space inclusions","Equivalent growth functions scramble Lebesgue exponents"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Orlicz exponents twist under equivalent Young functions","Trace Orlicz inclusion boils down to submultiplicativity","Logarithmic ratio stabilizes Orlicz space inclusions","Equivalent growth functions scramble Lebesgue exponents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"total_tokens":1358,"prompt_tokens":1116,"completion_tokens":242,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":732,"completion_tokens_details":{"reasoning_tokens":179}},"tokens_in":732,"tokens_out":242,"duration_ms":3680,"temperature":1.0,"reasoning_tokens":179,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:26:31.064799+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bonino, S","cited_arxiv_id":null,"evidence_quote":"Defines the Lebesgue exponents and supplies the comparison inequalities and equivalence results that Section 2 refines."},{"cited_title":"Maligranda, Indices and interpolation,Diss","cited_arxiv_id":null,"evidence_quote":"Provides the index bounds and limit-invariance arguments used in Proposition 2.1, Lemma 2.17, and related steps."},{"cited_title":"Maligranda, Calderón–Lozanovskii; construction for mixed norm spaces, Acta Math","cited_arxiv_id":null,"evidence_quote":"Gives the mixed-norm inclusion criteria on which Proposition 3.12 and the trace-space characterizations rest."},{"cited_title":"Finol, L","cited_arxiv_id":null,"evidence_quote":"Establishes that equality of $L^{\\Phi,\\Phi}$ and $L^\\Phi$ occurs only in the Lebesgue-space case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the submultiplicativity and supermultiplicativity framework and the classification of Young functions that are both."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Lebesgue exponents whose instability is the paper's point of departure."}],"review_version":1}