{"id":"ad597451-e4fb-4418-a77e-ed9db583b8d6","arxiv_id":"2501.08997","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Homogeneous Besov and Triebel-Lizorkin spaces are defined on arbitrary homogeneous groups for all p,q,sigma, proved independent of the Littlewood-Paley decomposition, and characterized by maximal functions and molecular decompositions.","lead":"This paper develops a full theory of homogeneous Besov and Triebel-Lizorkin spaces on nilpotent Lie groups equipped with dilations, for all smoothness and integrability parameters. A generalist should care because this gives one language in which Hardy, Sobolev, and Lipschitz spaces on these groups fit together, with Littlewood-Paley, maximal, and wavelet characterizations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.6 is not proved internally: it is a direct application of a to-appear memoir [71], and the hypotheses (L^r_w-compatibility of the mixed-norm spaces) are only asserted via skipped proofs of Lemmas 6.5/6.7 and a citation to [71, Cor. 3.9].","rationale":"The reader's verdict (CONDITIONAL) is well-founded. After reviewing the proofs of Theorems 4.2 and 5.1, they appear internally coherent and do not rely on [71]. The independence result rests on the almost-orthogonality estimate (Lemma 3.1), the sub-mean-value property (Lemma 3.5), and the vector-valued maximal inequalities (Lemmas 3.8-3.13); these proofs are detailed and seem correct. The existence of phi (Proposition 3.4) is a prerequisite, but its proof relies on two cited facts ([21, Thm 4.1], [40, Lemma 7.1]) that are not reproduced; this is a secondary concern. The single most load-bearing unresolved point is Theorem 7.6, which is a black-box application of [71]. The paper does not contain the proof of the molecular decomposition, and the compatibility verification is largely skipped. This matches the reader's weakest_assumption's second half. Since the central claim explicitly includes molecular decompositions, the paper should be CONDITIONAL until the compatibility conditions are checked. No internal contradiction was found in Sections 3-5; the issue is deferred verification, not a demonstrated error.","tokens_in":77503,"tokens_out":30190,"duration_ms":288264,"concrete_test":"Independently verify the hypotheses of [71, Theorem 6.14] for the specific spaces P^{p,q}_{a,sigma} and L^{p,q}_{a,sigma}. Concretely, compute the operator norms ||L_{(y,t)}|| and ||R_{(y,t)}|| directly from Definition 6.4/6.6 (performing the change of variables used in the skipped proof of Lemma 6.5), and check whether they match the stated formulas in Lemma 6.5 and (6.5)-(6.6). If the formulas are confirmed, then also check the remaining L^r_w-compatibility conditions in [71, Definition 3.5]. A mismatch in any displayed exponent would invalidate the control-weight construction and hence Theorem 7.6.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest_claim includes molecular frame decompositions (Theorem 7.6). That theorem is obtained by citing [71, Theorem 6.14]; no proof is given in the paper. For the citation to apply, the spaces Y^{p,q}_{a,sigma} (P^{p,q}_{a,sigma} and L^{p,q}_{a,sigma}) must be L^r_w-compatible with the standard control weight w of Lemma 7.2. The verification is deferred: Lemma 6.5 and Lemma 6.7 state the required solid quasi-Banach/r-norm properties and the operator-norm estimates, but their proofs are skipped ('very similar ... hence skipped'), and Lemma 7.2 only sketches the construction of w. If any of these estimates is wrong (e.g., the exponent in (6.5)-(6.6) or the factor max{1,t^Q} in Lemma 6.5 for p=infty), then w fails condition (c3) or (c4) of Definition 7.1, and [71, Theorem 6.14] cannot be invoked. The molecular decomposition claim would then be unproved, and the paper's central claim would not be fully established. This is a genuine soft spot because it is not an internal proof but a chain of unverified external dependencies (including a memoir by one of the authors listed as to appear).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Littlewood-Paley-type theory of homogeneous Besov spaces Ḃ^σ_{p,q}(N) and Triebel-Lizorkin spaces Ḟ^σ_{p,q}(N) on an arbitrary homogeneous group N, for all p,q∈(0,∞] and σ∈R. The spaces are defined using Schwartz functions with vanishing moments that satisfy discrete and continuous Calderón conditions. The main results are the independence of the spaces from the choice of Littlewood-Paley decomposition (Theorem 4.2), continuous maximal-function characterizations (Theorem 5.1), wavelet-transform characterizations (Section 6), molecular frame decompositions obtained through abstract coorbit theory (Theorem 7.6), and identifications with Hardy, BMO, Sobolev, and Lipschitz spaces (Section 8). The construction of suitable Calderón functions uses spectral multipliers of the non-differential homogeneous operator P defined in (3.9)–(3.10).","tokens_in":77746,"tokens_out":13305,"duration_ms":134148,"significance":"If the main theorems are correct, the paper achieves a genuinely decomposition-independent theory of homogeneous Besov and Triebel-Lizorkin spaces on arbitrary homogeneous groups, including the full quasi-Banach range and the case p=∞ for Triebel-Lizorkin spaces. The independence theorem and the continuous maximal characterizations are proved in considerable detail from first principles, and the identifications with Hardy, BMO, and Sobolev spaces are substantial and mostly self-contained. The main caveat is that the molecular decomposition theorem is imported from the to-appear memoir [71], and the verification of its hypotheses rests on lemmas whose proofs are skipped; Proposition 8.5 on Lipschitz spaces is also only sketched. These points do not undermine the core derivation of the independence and maximal-characterization results, but they need to be addressed before the molecular decomposition claims can be regarded as fully established.","major_comments":[{"comment":"These two lemmas are stated without proof, with the explanation that the proof is 'very similar' to Euclidean analogues and 'hence skipped.' The lemmas are load-bearing: they provide the solid quasi-Banach r-norm property and the translation operator-norm estimates that, together with [71, Cor. 3.9], establish the L^r_w-compatibility of the mixed-norm spaces P^{p,q}_{a,σ} and L^{p,q}_{a,σ}. This compatibility is exactly what is needed for the coorbit identification in Lemma 7.4 and hence for Theorem 7.6. Please provide complete proofs or give precise theorem numbers in [53,54,69] together with the explicit modifications required on homogeneous groups, including the p=∞ operator norm with the extra max{1,t^Q} factor and the exponents in (6.5)–(6.6).","section":"§6.2, Lemmas 6.5 and 6.7"},{"comment":"Theorem 7.6, which is advertised as a central result, is not proved internally: it is obtained by invoking [71, Theorem 6.14] from a to-appear memoir. The application requires the verification that the pair (Y^{p,q}_{a,σ'},w) is L^r_w-compatible in the sense of [71, Def. 3.5]. This verification is only sketched: it relies on the skipped Lemmas 6.5 and 6.7, on [71, Cor. 3.9], and on the construction of the control weight w in Lemma 7.2, which is described briefly. Please state precisely the theorem from [71] that is being applied, verify all of its hypotheses in the present setting, and either include the verification or cite a publicly accessible version of [71] with specific theorem numbers.","section":"§7.3, Theorem 7.6"},{"comment":"The identification of the Besov spaces Ḃ^σ_{∞,∞}(N) with homogeneous Lipschitz spaces is asserted in Proposition 8.5 with the sentence 'One may use the argument in [48], with minor modifications, to also prove a Littlewood-Paley characterization of the homogeneous spaces Λ̇^s(N).' No proof or detailed reference is supplied. Since this identification is part of the paper's advertised scope, please provide the full argument or a precise published reference that contains the homogeneous Littlewood-Paley characterization on stratified groups.","section":"§8.4, Proposition 8.5"}],"minor_comments":[{"comment":"The statement of Theorem 7.6 says that w is a standard control weight for Y^{p,q}_{a,σ}, but the proof and Lemma 7.4 use Y^{p,q}_{a,σ'} with σ' = σ+Q/2−Q/q. This looks like a typo and should be corrected.","section":"§7.3, Theorem 7.6"},{"comment":"The sentence 'Note that Lemma 3.8 is applicable provided that r < p∧q, which is satisfied precisely for a > ar/q∧q > Q/(p∧q)' contains a garbled inequality; it should say that such an r exists because a > Q/(p∧q).","section":"Proof of Theorem 4.2, Step 2"},{"comment":"In the computation of the isometry (6.2), the text says 'using that Vψf(x,t) = t^{Q/2}(f∗ψ_t^∨)(x) and that φ is real and even'; the reference to φ should be to ψ.","section":"§6.1"},{"comment":"The expression 'max{1,s^Q} p ∞' in the definition of v_1 is unclear; please define the dependence on p=∞ explicitly, for instance by writing max{1,s^Q}^{1_{p=∞}}.","section":"Lemma 7.2"}],"recommendation":"major_revision","confidential_remarks":"The core results on independence and continuous maximal characterizations are proved in detail and appear sound. The main risk is the dependence of the molecular decomposition results on the to-appear memoir [71], together with the skipped proofs of Lemmas 6.5 and 6.7 and the sketched proof of Proposition 8.5; these are not fatal in principle, but they mean that some advertised results are conditional on external work. If the memoir is not yet available in final form, the editorial decision should weigh whether the paper can be accepted with that dependence made fully explicit and with the missing verifications supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the core of this paper is a real contribution. It builds homogeneous Besov and Triebel-Lizorkin spaces on arbitrary homogeneous groups using Littlewood-Paley decompositions from the non-differential homogeneous operator P of Glowacki/Dziubanski, proves independence of the decomposition, gives continuous maximal characterizations, and slots Hardy, Sobolev, and BMO spaces into the scale. The new ground is genuine: no self-adjoint operator with Gaussian kernel estimates is needed, and the p=infty Triebel-Lizorkin case on stratified groups is new.\n\nThe analytic engine is Sections 3-5. Lemma 3.1 (almost orthogonality), Lemma 3.5 (sub-mean-value), and the vector-valued inequalities are proved in detail and look internally consistent. Theorem 4.2 (independence) and Theorem 5.1 (maximal characterizations) are the heart of the paper and are established from first principles with standard tools. Those results alone justify the paper.\n\nThe soft spot is Section 7. Theorem 7.6 is not proved internally; it is a direct application of [71, Theorem 6.14], a to-appear memoir by one of the authors. To invoke it, the mixed-norm spaces Y need to be L^r_w-compatible with the standard control weight. That compatibility rests on Lemmas 6.5 and 6.7, whose proofs are skipped, and on Lemma 7.2, which sketches the weight. The stress-test note is correct: if any of those operator-norm estimates is off, especially the p=infty case or the max{1,t^Q} factor, the weight fails conditions (c3)/(c4) and [71] cannot be applied. This is a genuine verification gap for the molecular/frame claims. It does not affect the independence or maximal characterizations, but the abstract-coorbit chain is load-bearing for the decomposition results.\n\nMinor: Proposition 8.5 (Lipschitz identification) is deferred to minor modifications of [48]. That is a small issue, not a structural one. The citation pattern is fine; reliance on [71] is disclosed, though its to-appear status makes those parts conditional.\n\nBottom line: this deserves a serious referee. I would send it to a good analysis journal and ask the authors to expand the skipped proofs in Section 6 or clearly state the dependency on [71]. The core theory is solid and new.","headline":"A careful, mostly self-contained development of Besov/Triebel-Lizorkin theory on arbitrary homogeneous groups; the independence and maximal characterizations hold up, but the molecular decomposition section leans on a to-appear memoir whose hypotheses are only asserted.","tokens_in":78318,"tokens_out":4378,"would_cite":true,"duration_ms":41269,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E25","22E30","43A80","46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Besov and Triebel-Lizorkin spaces are well-defined on every homogeneous group.","keywords":["Besov spaces","Triebel-Lizorkin spaces","homogeneous groups","Littlewood-Paley decomposition","Calderón reproducing formula","molecular frame decompositions","Hardy spaces on homogeneous groups","Sobolev spaces on graded Lie groups"],"falsifier":"A direct check of Proposition 3.4 would settle the construction: pick a homogeneous quasi-norm, take the kernel $k$ of $m(P)$ for a smooth compactly supported multiplier $m$, and test whether $k\\in\\mathcal{S}_0(N)$; if for some homogeneous group the kernel failed to have all moments vanishing, the Calderón functions would not exist as claimed. Alternatively, if two functions $\\varphi,\\eta\\in\\mathcal{S}_0(N)$ satisfying the discrete Calderón condition could be exhibited with inequivalent $\\dot{\\mathbf{B}}^{\\sigma}_{p,q}$ quasi-norms for some $p,q,\\sigma$, Theorem 4.2 would be false.","tokens_in":77274,"feed_emoji":"📐","tokens_out":7864,"duration_ms":71427,"temperature":0.7,"pith_summary":"The paper develops homogeneous Besov spaces $\\dot{\\mathbf{B}}^{\\sigma}_{p,q}(N)$ and Triebel-Lizorkin spaces $\\dot{\\mathbf{F}}^{\\sigma}_{p,q}(N)$ on an arbitrary homogeneous group $N$, for every smoothness exponent $\\sigma\\in\\mathbb{R}$ and all integrability exponents $p,q\\in(0,\\infty]$. Its central aim is to show that these spaces are genuine objects of harmonic analysis rather than artifacts of a chosen Littlewood-Paley decomposition: the definitions start from a Schwartz function $\\varphi\\in\\mathcal{S}_0(N)$ with all moments vanishing that satisfies a discrete Calderón reproducing formula, and Theorem 4.2 proves that different choices give the same spaces with equivalent quasi-norms. The paper also establishes continuous maximal-function characterizations for the full parameter range, molecular frame decompositions, and identifications of classical Hardy, Sobolev, and Lipschitz spaces on homogeneous groups as special cases. This matters because existing theories were tied to stratified or graded groups or required operators whose semigroups satisfy Gaussian estimates, whereas the kernels used here come from a non-differential homogeneous convolution operator available on every homogeneous group.","feed_headline":"Littlewood-Paley spaces are well-defined on all homogeneous groups","feed_subtitle":"New decomposition-independent scale includes Hardy, Sobolev, and Lipschitz spaces for all exponents and smoothness orders.","key_machinery":"The load-bearing mechanism is the non-differential homogeneous convolution operator $P$ defined through a singular integral with kernel $\\rho(x)^{-(Q+1)}$ on an arbitrary homogeneous group; $P$ is homogeneous of degree $1$, positive, and essentially self-adjoint. Spectral multipliers $m(P)$ of this operator have Schwartz convolution kernels with all moments vanishing, which yields the functions $\\varphi\\in\\mathcal{S}_0(N)$ satisfying both the discrete and continuous Calderón conditions (Proposition 3.4). The proofs then run on an almost-orthogonality estimate for convolution products of dilated Schwartz functions, a sub-mean-value property of the convolution products $f\\ast\\varphi_t$, Peetre-type maximal functions, vector-valued Hardy-Littlewood maximal inequalities, and, for the molecular decompositions, the realization of the spaces as coorbit spaces for the quasi-regular representation of the semidirect product $G=N\\rtimes(0,\\infty)$.","core_discovery":"On the paper's own terms, the central discovery is that a Littlewood-Paley-type decomposition on any homogeneous group, built from any $\\varphi\\in\\mathcal{S}_0(N)$ satisfying the discrete Calderón condition $f=\\sum_{j\\in\\mathbb{Z}} f\\ast\\varphi_{2^{-j}}\\ast\\varphi_{2^{-j}}$, defines Besov and Triebel-Lizorkin spaces whose (quasi-)norms are independent of the auxiliary function for all $p,q\\in(0,\\infty]$ and all $\\sigma\\in\\mathbb{R}$. Theorem 4.2 establishes this equivalence, including the previously untreated case of Triebel-Lizorkin spaces with $p=\\infty$, which is handled through dyadic-ball Carleson-type norms. Theorem 5.1 provides continuous Peetre-type maximal-function characterizations, and Theorem 7.6 produces molecular frame expansions $f=\\sum_{\\lambda\\in\\Lambda}\\langle f,\\varphi_\\lambda\\rangle_\\psi\\, \\pi(\\lambda)\\psi = \\sum_{\\lambda\\in\\Lambda}\\langle f,\\pi(\\lambda)\\psi\\rangle_\\psi\\, \\varphi_\\lambda$ valid for all elements of the spaces. As a consequence, zero-order operators built from homogeneous convolution kernels, such as $X^\\alpha P^{-[\\alpha]}$, are bounded on the whole scale.","pith_inferences":["Beyond the paper, the coorbit interpretation suggests that explicit atomic or wavelet-coefficient characterizations with concrete sequence spaces should hold, though the paper obtains the frame system through an abstract molecular theorem rather than by constructing a universal analyzing vector.","One testable next step is whether the scale has interpolation and duality properties analogous to Euclidean Besov and Triebel-Lizorkin spaces; the paper does not prove these.","The boundedness criterion for order-zero kernels likely applies to many singular integral operators on homogeneous groups, not only the examples $X^\\alpha P^{-[\\alpha]}$ treated here.","Because the $p=\\infty$ Triebel-Lizorkin case is new even on stratified groups, it would be instructive to check whether the dyadic-ball norm used here reproduces the expected BMO-type identification in that more classical setting."],"forward_implications":["For any homogeneous group, the Besov and Triebel-Lizorkin spaces are independent of the choice of Littlewood-Paley decomposition, so the scale is intrinsic to the group rather than to a chosen kernel.","Continuous maximal-function characterizations hold for the full range of parameters $p,q\\in(0,\\infty]$, $\\sigma\\in\\mathbb{R}$, including the Triebel-Lizorkin spaces at $p=\\infty$.","Molecular frame decompositions exist, yielding expansions in terms of $\\pi(\\lambda)\\psi$ and a molecular dual system, with convergence in the weak-$*$ topology of $\\mathcal{S}'_0(N)$.","Zero-order convolution operators with kernels that are smooth away from the identity and homogeneous of degree $-Q$ are bounded on every space in the scale.","Classical Hardy spaces, homogeneous Sobolev spaces associated with Rockland operators, and Lipschitz spaces on stratified groups appear as special cases of these spaces."],"supporting_citations":[{"why":"Supplies the theory of homogeneous groups and Hardy spaces, including vector-valued singular integrals and commutative approximate identities, used in the definitions and in Section 8.","marker":"[29]"},{"why":"Constructs the non-differential homogeneous convolution operator $P$ and stable semigroups used to produce Calderón functions on arbitrary homogeneous groups.","marker":"[38]"},{"why":"Proves that spectral multipliers of $P$ have Schwartz convolution kernels, a key step in Proposition 3.4.","marker":"[21]"},{"why":"Provides the symbolic-calculus argument that these convolution kernels have all moments vanishing, placing them in $\\mathcal{S}_0(N)$.","marker":"[40]"},{"why":"The earlier homogeneous Besov space theory on stratified Lie groups whose Calderón and wavelet machinery is generalized to arbitrary homogeneous groups.","marker":"[32]"},{"why":"The earlier homogeneous Triebel-Lizorkin space theory on stratified groups, which is a special case of the present framework.","marker":"[47]"},{"why":"The abstract coorbit decomposition theorem that supplies the molecular frame expansions used in Theorems 7.6 and 7.8.","marker":"[71]"},{"why":"Defines homogeneous Sobolev spaces on graded Lie groups via Rockland operators, identified with the Triebel-Lizorkin spaces in Proposition 8.3.","marker":"[26]"}],"fun_headline_variants":["Homogeneous groups host full Besov–Triebel scale","Decomposition independence proven for homogeneous group spaces","Unified Besov/Triebel spaces for all exponents on homogeneous groups","From Hardy to Lipschitz: one scale on homogeneous groups","Littlewood-Paley spaces on homogeneous groups: no choice needed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the existence of a Schwartz function $\\varphi\\in\\mathcal{S}_0(N)$ satisfying the discrete and continuous Calderón reproducing formulas on every homogeneous group; the proof obtains it by spectral calculus of the non-differential operator $P$, relying on cited facts about Schwartz kernels and vanishing moments, and the molecular decomposition additionally assumes that an abstract coorbit decomposition theorem applies to the resulting weighted spaces.","fun_headline_variants_meta":{"raw":{"variants":["Homogeneous groups host full Besov–Triebel scale","Decomposition independence proven for homogeneous group spaces","Unified Besov/Triebel spaces for all exponents on homogeneous groups","From Hardy to Lipschitz: one scale on homogeneous groups","Littlewood-Paley spaces on homogeneous groups: no choice needed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1413,"prompt_tokens":943,"completion_tokens":470,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":386}},"tokens_in":559,"tokens_out":470,"duration_ms":4893,"temperature":1.0,"reasoning_tokens":386,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:13:26.697261+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check of Proposition 3.4 would settle the construction: pick a homogeneous quasi-norm, take the kernel $k$ of $m(P)$ for a smooth compactly supported multiplier $m$, and test whether $k\\in\\mathcal{S}_0(N)$; if for some homogeneous group the kernel failed to have all moments vanishing, the Calderón functions would not exist as claimed. Alternatively, if two functions $\\varphi,\\eta\\in\\mathcal{S}_0(N)$ satisfying the discrete Calderón condition could be exhibited with inequivalent $\\dot{\\mathbf{B}}^{\\sigma}_{p,q}$ quasi-norms for some $p,q,\\sigma$, Theorem 4.2 would be false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theory of homogeneous groups and Hardy spaces, including vector-valued singular integrals and commutative approximate identities, used in the definitions and in Section 8."},{"cited_title":"G/suppress lowacki","cited_arxiv_id":null,"evidence_quote":"Constructs the non-differential homogeneous convolution operator $P$ and stable semigroups used to produce Calderón functions on arbitrary homogeneous groups."},{"cited_title":"Dziuba´ nski","cited_arxiv_id":null,"evidence_quote":"Proves that spectral multipliers of $P$ have Schwartz convolution kernels, a key step in Proposition 3.4."},{"cited_title":"G/suppress lowacki.Lp-boundedness of flag kernels on homogeneous groups via symbo lic calculus","cited_arxiv_id":null,"evidence_quote":"Provides the symbolic-calculus argument that these convolution kernels have all moments vanishing, placing them in $\\mathcal{S}_0(N)$."},{"cited_title":"F¨ uhr and A","cited_arxiv_id":null,"evidence_quote":"The earlier homogeneous Besov space theory on stratified Lie groups whose Calderón and wavelet machinery is generalized to arbitrary homogeneous groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The earlier homogeneous Triebel-Lizorkin space theory on stratified groups, which is a special case of the present framework."},{"cited_title":"Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition","cited_arxiv_id":"2203.07959","evidence_quote":"The abstract coorbit decomposition theorem that supplies the molecular frame expansions used in Theorems 7.6 and 7.8."},{"cited_title":"Fischer and M","cited_arxiv_id":null,"evidence_quote":"Defines homogeneous Sobolev spaces on graded Lie groups via Rockland operators, identified with the Triebel-Lizorkin spaces in Proposition 8.3."}],"review_version":1}