{"id":"aef2814a-b323-43b8-9e02-d208a4bf6cb5","arxiv_id":"2608.06813","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp fractional Gagliardo-Nirenberg inequalities in ball Banach function spaces have optimal rearrangement invariant target spaces, namely the Calderón-Lozanovskii spaces, with sharp constants as the smoothness index approaches 0 or 1.","lead":"Harmonic analysts found the exact optimal function spaces for a family of fractional interpolation inequalities, including endpoint cases with bounded mean oscillation. The result resolves an open question about which target spaces are both sufficient and necessary in a very general setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified to the optimal target-space characterization; the reader's BBM density concern affects Theorem 1.5, not Theorem 1.9.","rationale":"The reader's weakest assumption is about simultaneous density of C_c^∞ in X∩W^{k,Y} for the BBM limit (Theorem 1.5). That is a genuine technical concern for that theorem, but it is structurally distinct from the central optimal-target claim in Theorem 1.9. The proof of Theorem 1.9 uses Theorem 1.1 for sufficiency and Theorem 4.1 for necessity; neither depends on the BBM density. I therefore examined the separated bump construction in Theorem 4.1 carefully. The norm comparisons of f_{R,L} with u and v rely only on rearrangement-invariant norm properties and the dilation bound (4.3), with constants independent of R and L. The error estimate (4.19) is obtained by restricting to |h|>L/(2k), where the integrand is supported on a set of h of measure bounded by the volumes of the displaced balls; this yields the decay L^{−n/q−sk}, which vanishes as L→∞. The resulting liminf lower bound (4.12) is therefore justified, and after applying the assumed inequality (4.4) the R^{−sk} factors cancel, giving (4.7) with a constant independent of the simple functions. I also spot-checked the exponent bookkeeping in Proposition 2.1, the main technical engine for sufficiency, and found the geometric series convergent under the stated choice of ε. Since the central characterization appears internally consistent and its proof does not rely on the BBM density step, I have no significant objection to the central claim. The BBM density issue, while real, would at most require a separate justification for Theorem 1.5 and does not change the verdict on the optimal target-space characterization.","tokens_in":52502,"tokens_out":36564,"duration_ms":294993,"concrete_test":"As a worthwhile verification, independently re-derive the lower-bound claim (4.12) for the separated bump construction with a concrete instance: take n=1, k=1, s=1/2, q=2, X=Y=L^2, and u=v=1_{(0,1)}. Compute ∥D^{1/2,1}_2(f_{R,L})∥_{L^2} directly and check that the liminf as L→∞ is at least c R^{−1/2}∥1_{(0,1)}∥_{L^2} with c independent of R and L; this exercises the key step on which the necessity direction of Theorem 1.9 rests.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 1.9, asserts that X^{1-s}Y^s is both sufficient and necessary among rearrangement invariant targets. Sufficiency follows from the pointwise estimates in Theorem 1.1 and maximal boundedness; necessity is proved in Theorem 4.1 via a separated bump construction. I traced the construction: the distribution comparisons (4.15)–(4.17) use only rearrangement invariance and the dilation bound (4.3) with absolute constants; the exponent balance in Proposition 2.1 Case 2 is consistent (the geometric factor 2^{jq(ε−s0k)+jn(θ−1)_+} is summable because ε<δ0/2); and the cross-term error in (4.19) decays as L^{−n/q−sk}, so the liminf lower bound (4.12) is justified. The constants cancel with R^{−sk}, yielding (4.7) with a uniform constant. I found no internal inconsistency or missing hypothesis in this argument. The reader's flagged assumption about simultaneous density of C_c^∞ in X∩W^{k,Y} is a legitimate concern for the BBM formula (Theorem 1.5), but that theorem is not used in the optimal-target proof; thus it is not load-bearing for the central characterization.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a fractional Gagliardo–Nirenberg theory in the framework of ball Banach function spaces. The main technical input is a pair of pointwise estimates for the k-th order fractional difference operator D^{s,k}_q(f), with constants tracking the dependence on s∈(0,1): Theorem 1.1(i) uses the sharp maximal function and a powered Hardy–Littlewood maximal function, while Theorem 1.1(ii) extends the Maz'ya–Shaposhnikova pointwise estimate to higher order and off-diagonal parameters. These pointwise estimates are then converted into norm inequalities on Calderón–Lozanovskii spaces X^{1-s}Y^s (Theorem 1.3), including a BMO endpoint. The paper also proves Bourgain–Brezis–Mironescu limits as s→1⁻ (Theorem 1.5) and Maz'ya–Shaposhnikova type asymptotic bounds as s→0⁺ (Theorem 1.7), showing that the asymptotic factors in Theorem 1.3 have optimal order. The central result is Theorem 1.9: for fixed s, under a sharp parameter condition, the inequality with target B holds for all f in the appropriate intersection if and only if X^{1-s}Y^s embeds into B, when X,Y,B are rearrangement invariant Banach function spaces. The necessity is proved in Theorem 4.1 by a separated-bump construction that avoids the additional assumption X⊂Y imposed in earlier work.","tokens_in":52662,"tokens_out":17112,"duration_ms":146508,"significance":"If correct, the paper settles a natural and nontrivial optimality question: the Calderón–Lozanovskiı space is not merely an admissible target but the minimal rearrangement invariant target for fractional Gagliardo–Nirenberg inequalities. The separated-bump necessity proof is a genuine improvement over the prior X⊂Y-constrained argument of [57, Corollary 2.4], and it yields a new integer-order optimality statement as well. The pointwise estimates with explicit s-dependence are likely to be useful beyond the present applications, and the BMO endpoint and off-diagonal results are new in several of the function-space scales treated in Section 6. The paper is well structured: Theorem 1.1 is proved from dyadic telescoping and covering estimates, Theorem 1.9(i) combines those estimates with maximal boundedness, and the optimality direction is self-contained up to standard rearrangement-invariant space facts. The parameter sharpness in Section 5 is also explicitly tested by constructed examples. I found no internal contradiction in the main argument, and I am not aware of an error that would undermine the central characterization.","major_comments":[],"minor_comments":[{"comment":"The proof of Theorem 1.5 uses the simultaneous C_c^∞-density of X∩W^{k,Y} in both norms, imported from [17, Corollary 3.10]. Please state the precise density result and its hypotheses explicitly, since this is the one ingredient I did not independently verify; this concern does not affect Theorem 1.9 or Theorem 4.1.","section":"§3.1, Eq. (3.8), Proposition 3.2"},{"comment":"The word 'Corrollary' appears twice in the discussion of the proof strategy for Theorem 1.9 and should be corrected to 'Corollary'.","section":"§1, proof of Theorem 1.9; §4, proof of Theorem 4.1"},{"comment":"The optimality statement in Theorem 1.9 changes the status of X and Y from general ball Banach function spaces in the inequality part to rearrangement invariant Banach function spaces in the if-and-only-if part; this should be made explicit in the theorem statement to avoid confusion.","section":"§1, Theorem 1.9"},{"comment":"In the necessity part of Theorem 6.2, the scaling argument is only sketched; a one-line display showing that the scaling exponents force er=r would improve readability.","section":"§6.2, Theorem 6.2"},{"comment":"In the definition of δ0 in the proof of Proposition 2.1(i), the displayed formula would be clearer if the parentheses and the positive-part notation were typeset unambiguously, since the subsequent summability argument depends on the exact relation between δ0 and the exponent s0k - n((1-s0)/p1 + s0/p2 - 1/q).","section":"§2, Proposition 2.1"}],"recommendation":"accept","confidential_remarks":"This is a substantial paper with a clear main claim and a convincing proof structure. The only point I did not fully verify is the imported simultaneous density result used in the BBM limit theorem, but it is a literature citation and does not affect the central optimal-target characterization. The paper is well within the scope of the journal and the novelty is clearly articulated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe short version: this is a real paper and the main claim holds up. The authors prove that for rearrangement invariant spaces the Calderón–Lozanovskii space X^{1-s}Y^s is not just an admissible target but the minimal one, and they answer the Lorentz-space question from Leśnik–Roskovec–Soudský. I found no circularity in the optimality argument: the bump construction forces the embedding directly, and the exponent bookkeeping checks out.\n\nWhat is genuinely new: two pointwise fractional difference estimates with explicit s-dependence (Theorem 1.1), norm inequalities including BMO endpoints and off-diagonal cases (Theorems 1.3 and 1.9), sharp BBM/MS asymptotics, and the if-and-only-if target characterization. The proofs are mostly self-contained where it matters. The necessity direction in Theorem 4.1 uses separated bumps and rearrangement invariance only; I traced (4.15)–(4.17) and the cross-term decay in (4.19), and it works. The sharpness counterexamples in Section 5 are honest and test the right parameter ranges.\n\nSoft spots, in proportion. The BBM limit (Theorem 1.5) imports a simultaneous density of C_c^∞ in X∩W^{k,Y} from [17, Corollary 3.10]. That is a real assumption, and if it fails for some admissible ball Banach function spaces the formula is only proved for test functions. But it does not touch the optimal-target theorem, which is the center of the paper. Endpoint p=1 spaces are excluded; the authors say so plainly and point to a different sparse technique needed. The paper leans heavily on earlier work by the same group, but those are published tools with independent proofs, so I do not see this as a flaw. I did not verify every estimate against the 100+ references; confidence is moderate rather than high, mainly because the technical scaffolding is dense.\n\nWho gets value: anyone working in interpolation, Sobolev embeddings, or rearrangement invariant spaces. The applications to Lorentz and Orlicz scales are concrete, and the answer to the open question is the kind of thing people will cite.\n\nRecommendation: send it to referees. It deserves a serious referee; I expect heavy but manageable revisions, mostly around making the density hypothesis in Theorem 1.5 more transparent or isolating it as an assumption rather than a black box.","headline":"Solid, technically dense paper; the optimal target-space characterization is new and the necessity argument checks out, with the only real caveat being an imported density assumption in the BBM limit theorem.","tokens_in":53280,"tokens_out":1918,"would_cite":true,"duration_ms":18707,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","46E35","42B25","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the optimal rearrangement-invariant target space for fractional Gagliardo–Nirenberg inequalities is precisely the Calderón–Lozanovskii construction, settling an open Lorentz-space question.","keywords":["fractional Gagliardo–Nirenberg inequality","pointwise estimate","sharp asymptotics","optimal target space","Calderón–Lozanovskii space","ball Banach function space","rearrangement invariant space","BMO endpoint"],"falsifier":"Compute the ratio $\\|D^{s,k}_q(f_{R,L})\\|_B/(\\|f_{R,L}\\|_X^{1-s}\\|\\nabla^k f_{R,L}\\|_Y^s)$ for the separated-bump functions of (4.11) while letting $L\\to\\infty$; the theorem predicts the liminf is bounded below by $\\|u^{1-s}v^s\\|_B$, so a pair of rearrangement-invariant spaces where this lower bound fails would refute the optimality claim. For the sharpness of the parameter condition, the function $\\varphi$ of Lemma 5.1 satisfies $D^{s,k}_q(\\varphi)(x)\\gtrsim |x|^{-n/q-sk}$, and the divergence of $\\int^\\infty |x|^{-p_s(n/q+sk)}\\,dx$ when $n(\\frac1{p_s}-\\frac1q)\\ge sk$, with $1/p_s=(1-s)/p_1+s/p_2$, already shows no inequality can hold outside the stated range.","tokens_in":52221,"feed_emoji":"🎯","tokens_out":10928,"duration_ms":81948,"temperature":0.7,"pith_summary":"Fractional Gagliardo–Nirenberg inequalities interpolate between a function and its $k$-th order gradient in different function spaces, with a fractional difference operator of order $s$ as the interpolated quantity. This paper builds a complete such theory inside ball Banach function spaces: it proves pointwise bounds with sharp dependence on $s$, derives norm inequalities including a BMO endpoint, and determines the exact endpoint asymptotics as $s\\to1^-$ and $s\\to0^+$. Its central discovery is optimality: when the underlying spaces are rearrangement invariant, an inequality with any rearrangement-invariant target $B$ holds for all $f$ if and only if the Calderón–Lozanovskii space $X^{1-s}Y^s$ embeds into $B$. This makes $X^{1-s}Y^s$ the minimal admissible target and answers an open question of Leśnik, Roskovec, and Soudský about Lorentz targets.","feed_headline":"Calderón–Lozanovskii spaces are the exact optimal targets","feed_subtitle":"New proof pins down the minimal rearrangement-invariant target and settles an open Lorentz-space question.","key_machinery":"The load-bearing object is the Calderón–Lozanovskii space $X^{1-s}Y^s$, defined as the set of functions pointwise dominated by $|f|^{1-s}|g|^s$ with norm $\\inf\\|f\\|_X^{1-s}\\|g\\|_Y^s$; it is simultaneously the natural interpolation target and, by the converse argument, the necessary target. The proof engine for the norm inequalities is a pair of pointwise estimates for the fractional difference operator $D^{s,k}_q(f)$, obtained by expanding $\\Delta^k_h f$ through shifted dyadic grids and minimizing polynomials (Lemma 2.4), with constants that scale like $[(s-s_0)(\\tilde s_0-s)]^{-1/q}$ and $[s(1-s)]^{-1/q}$. For the optimality direction, a new family of test functions combines separated localized bumps whose supports are far apart, so that rearrangement-invariant norms split into those of the underlying simple functions and force $u^{1-s}v^s$ into the target $B$.","core_discovery":"The central claim is Theorem 1.9. For fixed $s\\in(0,1)$, $k\\in\\mathbb{N}$, and $q\\in[1,\\infty)$, if the lower generalized Boyd indices satisfy $n\\bigl(\\frac{1-s}{p_X}+\\frac{s}{p_Y}-\\frac1q\\bigr)<sk$, then $$\\left\\|$D^{{s,k}}$_q(f)\\right\\|_{$X^{{1-s}}$Y^s}\\lesssim \\|f\\|$_X^{{1-s}}$\\|\\nabla^k f\\|_Y^s$$ for every $f\\in X\\cap\\dot{W}^{k,Y}$. Conversely, whenever $X,Y,B$ are rearrangement-invariant Banach function spaces and the same inequality holds with target $B$ for all such $f$, necessarily $X^{1-s}Y^s\\hookrightarrow B$; the BMO endpoint version forces $Y^{1/s}\\hookrightarrow B$. The Calderón–Lozanovskii space is therefore not one admissible target among many but the unique minimal rearrangement-invariant target. The same separated-bump argument removes the extra assumption $X\\subset Y$ from the integer-order optimality result, extends it to all $1\\le j<k$, and yields the explicit Lorentz-space characterization: the inequality is valid for target $L^{\\tilde r,\\tilde\\mu}$ iff $\\tilde r=r$ and $\\tilde\\mu\\ge\\mu$.","pith_inferences":["If the characterization is correct, computing the sharp constant in any concrete rearrangement-invariant instance reduces to computing the embedding constant of $X^{1-s}Y^s$ into the target; this converts a family of inequalities into a single embedding problem.","The separated-bump construction is a transferable template: any functional that is subadditive across widely separated bumps and satisfies a lower bound like (4.9) should admit the same minimal-target characterization among rearrangement-invariant spaces.","A natural testable extension is to replace rearrangement invariance by a weaker symmetry (for example, mixed-norm or variable Lebesgue spaces), where Theorem 1.9 gives the inequality but leaves the optimality question open.","The BMO endpoint suggests that other critical endpoint spaces (for instance exponential-type Orlicz spaces) could replace $L^\\infty$ in the lower-order factor, with the same Calderón–Lozanovskii product governing the target."],"forward_implications":["For Lorentz scales, the characterization is explicit: the fractional or integer Gagliardo–Nirenberg inequality with a Lorentz target holds exactly when the first index is the Calderón–Lozanovskii interpolation of the two source indices and the second index is at least the interpolated one; the case $j=1$, $k=2$ answers [57, Question 2.6].","The asymptotic factors in the inequalities are optimal: the norms diverge like $(1-s)^{-1/q}$ as $s\\to1^-$ and like $s^{-1/q}$ as $s\\to0^+$, so no uniform-in-$s$ constant can improve them.","Off-diagonal inequalities ($X\\neq Y$) and the BMO endpoint estimates are new for weighted Lebesgue, Morrey, Bourgain–Morrey, Lorentz, and Orlicz spaces.","The integer-order optimality theorem now covers all $1\\le j<k$ and removes the additional inclusion assumption $X\\subset Y$ used in the earlier optimality proof.","When the derivative space is a Lebesgue space, the BMO endpoint estimate sharpens to a double-integral bound with factor $1/((1-s)(sp-1))$, new for $k\\ge2$."],"supporting_citations":[{"why":"poses the open Lorentz-target question and contains the earlier integer-order optimality result under the additional assumption $X\\subset Y$ that the paper removes and answers.","marker":"[57]"},{"why":"provides the integer-order rearrangement-invariant Gagliardo–Nirenberg inequality used for the sufficiency direction in the Lorentz and Orlicz characterizations.","marker":"[58]"},{"why":"first established the Calderón–Lozanovskii space as the target for integer-order Gagliardo–Nirenberg inequalities, the pattern Theorem 1.9 extends and sharpens.","marker":"[24]"},{"why":"supplies the ball-Banach Sobolev-space framework and the simultaneous smooth-function density result on which the Bourgain–Brezis–Mironescu proof leans.","marker":"[17]"},{"why":"provides the localization and shifted-grid techniques that the pointwise estimates refine, as well as prior diagonal results used for comparison.","marker":"[40]"},{"why":"origin of the pointwise maximal-function Gagliardo–Nirenberg estimate that Theorem 1.1 generalizes to fractional differences.","marker":"[68]"},{"why":"gives the fractional pointwise estimate (1.2) that Theorem 1.1(ii) recovers when $k=1$.","marker":"[69]"},{"why":"states the Maz'ya–Shaposhnikova formula whose sharp order as $s\\to0^+$ Theorem 1.7 extends to Calderón–Lozanovskii norms.","marker":"[70]"},{"why":"previous ball-Banach Gagliardo-representation and Maz'ya–Shaposhnikova results that Theorem 1.7 improves.","marker":"[82]"},{"why":"the classical fractional Gagliardo–Nirenberg inequality on Lebesgue spaces recovered as a special case of Theorem 1.9.","marker":"[9]"}],"fun_headline_variants":["Calderón–Lozanovskii spaces are the unique optimal targets","Sharp Gagliardo–Nirenberg targets: a complete answer","Open problem solved: optimal G-N spaces characterized","BMO and off-diagonal cases settle the G-N target space"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Bourgain–Brezis–Mironescu limit in Theorem 1.5 imports the simultaneous density of smooth compactly supported functions in $X\\cap W^{k,Y}$ for both norms from a cited result; if this density failed for some admissible ball Banach function space, the endpoint formula would only be known for smooth functions and the passage to all $f$ would break.","fun_headline_variants_meta":{"raw":{"variants":["Calderón–Lozanovskii spaces are the unique optimal targets","Sharp Gagliardo–Nirenberg targets: a complete answer","Open problem solved: optimal G-N spaces characterized","BMO and off-diagonal cases settle the G-N target space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000291,"raw_usage":{"total_tokens":1747,"prompt_tokens":1037,"completion_tokens":710,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":653,"completion_tokens_details":{"reasoning_tokens":637}},"tokens_in":653,"tokens_out":710,"duration_ms":6399,"temperature":1.0,"reasoning_tokens":637,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:29:54.781000+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the ratio $\\|D^{s,k}_q(f_{R,L})\\|_B/(\\|f_{R,L}\\|_X^{1-s}\\|\\nabla^k f_{R,L}\\|_Y^s)$ for the separated-bump functions of (4.11) while letting $L\\to\\infty$; the theorem predicts the liminf is bounded below by $\\|u^{1-s}v^s\\|_B$, so a pair of rearrangement-invariant spaces where this lower bound fails would refute the optimality claim. For the sharpness of the parameter condition, the function $\\varphi$ of Lemma 5.1 satisfies $D^{s,k}_q(\\varphi)(x)\\gtrsim |x|^{-n/q-sk}$, and the divergence of $\\int^\\infty |x|^{-p_s(n/q+sk)}\\,dx$ when $n(\\frac1{p_s}-\\frac1q)\\ge sk$, with $1/p_s=(1-s)/p_1+s/p_2$, already shows no inequality can hold outside the stated range.","supporting_citations":[{"cited_title":"Le ´snik, T","cited_arxiv_id":null,"evidence_quote":"poses the open Lorentz-target question and contains the earlier integer-order optimality result under the additional assumption $X\\subset Y$ that the paper removes and answers."},{"cited_title":"Le ´snik, T","cited_arxiv_id":null,"evidence_quote":"provides the integer-order rearrangement-invariant Gagliardo–Nirenberg inequality used for the sufficiency direction in the Lorentz and Orlicz characterizations."},{"cited_title":"Fiorenza, M","cited_arxiv_id":null,"evidence_quote":"first established the Calderón–Lozanovskii space as the target for integer-order Gagliardo–Nirenberg inequalities, the pattern Theorem 1.9 extends and sharpens."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the localization and shifted-grid techniques that the pointwise estimates refine, as well as prior diagonal results used for comparison."},{"cited_title":"Maz’ya and T","cited_arxiv_id":null,"evidence_quote":"origin of the pointwise maximal-function Gagliardo–Nirenberg estimate that Theorem 1.1 generalizes to fractional differences."},{"cited_title":"Maz’ya and T","cited_arxiv_id":null,"evidence_quote":"gives the fractional pointwise estimate (1.2) that Theorem 1.1(ii) recovers when $k=1$."},{"cited_title":"Maz’ya and T","cited_arxiv_id":null,"evidence_quote":"states the Maz'ya–Shaposhnikova formula whose sharp order as $s\\to0^+$ Theorem 1.7 extends to Calderón–Lozanovskii norms."}],"review_version":1}