{"id":"c693eb9a-0ee4-4ab9-a572-c9514108f662","arxiv_id":"2505.19135","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines weighted homogeneous Bourgain-Morrey-Besov and Triebel-Lizorkin type spaces associated with an operator L and proves Peetre maximal, heat kernel, atomic, and molecular characterizations plus boundedness of fractional powers and spectral multipliers.","lead":"This paper introduces new weighted function spaces, Bourgain-Morrey-Besov and Triebel-Lizorkin type spaces, tied to a nonnegative self-adjoint operator with Gaussian heat kernel bounds, and proves their characterizations, atomic and molecular decompositions, and boundedness of fractional powers and spectral multipliers. It extends the authors' previous work and the Bui-Bui-Duong framework to these 'type' spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 3.9 reduces to the case q=2 by a substitution that does not preserve the norm; for q≠2 the Lusin/Littlewood-Paley characterizations in Theorem 3.10 are not established for the stated range.","rationale":"I read the paper as establishing a conditional theory: under the reverse-doubling and parameter restrictions, the spaces are defined and various norm equivalences are claimed. The reader's weakest assumption about the reverse-doubling condition and the dyadic-system dependence is legitimate, but both are assumptions or limitations that the theorem statements explicitly carry. The more concrete load-bearing problem I found is internal to the proof of Theorem 3.9. The reduction to q=2 does not preserve the Bourgain-Morrey norm: after the substitution, one either controls the wrong integrability exponent or the wrong power of the Lusin function, and no raising of the whole inequality can recover the displayed norm. Since Theorem 3.10 and Corollary 3.2 depend on Theorem 3.9 for all q in (0,p], the Lusin-function and Littlewood-Paley characterizations are not fully proved as written. This does not by itself invalidate the Peetre, heat-kernel, atomic, or molecular results, which is why I would keep the reader's CONDITIONAL verdict rather than move to REJECT: the gap is localized and likely repairable by a direct proof for general q. I would ask the authors to either supply a valid argument for q\\neq 2 or explicitly restrict Theorem 3.10 to the case q=2.","tokens_in":48061,"tokens_out":25527,"duration_ms":237819,"concrete_test":"Choose q=3, p=4 and follow the substitution in the proof of Theorem 3.9: set \\tilde F=(\\alpha^{-s}|F|)^{3/2}. Apply the q=2 estimate to \\tilde F with exponent P=8/3 and write out the resulting norm as (\\sum_Q \\omega(Q)^{r/t-r/P}(\\int_Q(S^{s,Q}_{a,3}F)^4 \\omega)^{r/P})^{1/r}. Since P=8/3\\neq p=4, the governed norm is not the claimed M^{t,r}_{4,\\omega} norm; if instead the q=2 estimate is applied with exponent 4, the integrand becomes (S^{s,Q}_{a,3}F)^6. No choice of exponent gives the displayed inequality. This verifies that the reduction is invalid for q\\neq 2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.5, Theorem 3.9: the proof says it suffices to prove the case s=0, q=2, and for general s,q sets \\tilde F(y,\\alpha)=(\\alpha^{-s}|F(y,\\alpha)|)^{q/2}. This gives S^{s,Q}_{a,q}F=(S^{0,Q}_{a,2}\\tilde F)^{2/q}, not equality of the quantities whose L^p norms are compared. Applying the q=2 estimate to \\tilde F with exponent P=2p/q controls the norm (\\sum_Q \\omega(Q)^{r/t-r/P} (\\int_Q (S^{s,Q}_{a,q}F)^p \\omega)^{r/P})^{1/r}, i.e. a Bourgain-Morrey norm with integrability P=2p/q rather than p; the outer exponent r/P differs from r/p. Applying it with exponent p instead controls (S^{s,Q}_{a,q}F)^{pq/2} inside the integral. Neither choice yields the stated inequality, and raising the whole inequality to a power changes the summation over Q in a way that cannot recover the claimed M^{t,r}_{p,\\omega} norm. Hence Theorem 3.9 is proved only for q=2, and Corollary 3.2 and Theorem 3.10, which invoke it for all 0<q\\le p<\\infty, are unsupported for q\\neq 2. This is an internal gap in the argument, not merely a parameter restriction; it affects the advertised Lusin-function and Littlewood-Paley characterizations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces associated with a nonnegative self-adjoint operator L on a space of homogeneous type satisfying the doubling property, reverse doubling condition, and infinite measure. The main results include characterizations via Peetre maximal functions (Theorem 3.5), compactly supported and noncompactly supported functional calculus (Theorems 3.7 and 3.8), heat-kernel characterizations (Corollary 3.1), Lusin-function and Littlewood-Paley characterizations (Theorems 3.9 and 3.10), atomic and molecular decompositions (Theorems 4.1-4.6), embeddings and completeness (Propositions 4.1-4.2 and Theorem 4.7), and boundedness of fractional powers and spectral multipliers (Theorems 5.1-5.2). The framework follows the pattern of Bui-Bui-Duong [20] and the earlier Bourgain-Morrey spaces of Bai-Xu [3], with the new feature that the norms are defined with cube-dependent truncations j >= j_Q.","tokens_in":48367,"tokens_out":10662,"duration_ms":98924,"significance":"If the main theorems are correct, the paper provides a useful extension of weighted Besov and Triebel-Lizorkin type spaces to the Bourgain-Morrey setting, with a full suite of real-variable characterizations and decompositions. The paper is transparently built on established machinery; many technical lemmas are quoted from [20] and [3], and the new definitions reduce to the non-type spaces of [3] when the cube truncation is removed, which is not circular but does mean the incremental novelty lies in the type-space norms. The manuscript contains machine-checkable statements in the sense that the proofs follow standard patterns, and the parameter restrictions such as r > -nt/log beta are explicitly stated. However, the advertised Lusin-function and Littlewood-Paley characterizations rely on Theorem 3.9, whose proof contains a gap for q != 2; this is a load-bearing issue that must be fixed before the main claims can be accepted.","major_comments":[{"comment":"The reduction to the case s=0 and q=2 is not valid as written. With tilde F(y,alpha) = (alpha^{-s}|F(y,alpha)|)^{q/2}, one has S^{0,Q}_{a,2} tilde F(x) = (S^{s,Q}_{a,q} F(x))^{q/2}. Substituting this into the q=2 estimate controls a weighted L^{pq/2}_omega norm, not the L^p_omega norm appearing in the statement; the outer exponent r/p and the summation over Q cannot be adjusted by raising powers without changing the quantity whose norm is estimated. Moreover, the proof of the q=2 case works only through a duality argument for p >= 2, and no argument is given for p < 2. Consequently Theorem 3.9 is established only for q=2, p >= 2, and Corollary 3.2 and Theorem 3.10, which invoke Theorem 3.9 for all 0 < q <= p < infinity, are unsupported for q != 2. This is an internal gap in the argument, not a mere parameter restriction, and it affects the advertised Lusin-function and Littlewood-Paley characterizations.","section":"Section 3.5, Theorem 3.9"},{"comment":"The spaces are defined with respect to a fixed dyadic cube system D chosen in Remark 2.1, and the norms depend essentially on the cube levels j_Q and on the collection D through the factors omega(Q)^{r/t - r/p}. The paper does not prove that the resulting spaces are independent of the choice of D up to equivalent norms. Lemma 3.3 only establishes independence for the Bourgain-Morrey norm M^{t,r}_{p,omega} over the adjacent systems, not for the truncated sums over j >= j_Q in Definitions 3.3 and 3.5, and Theorem 3.6 addresses only the choice of the partition of unity psi. Since the definitions are presented as defining 'the' weighted homogeneous Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces, the authors should either prove equivalence over the finite adjacent family of dyadic systems or state explicitly that the spaces depend on the chosen system D.","section":"Definitions 3.3 and 3.5 and subsequent theorems"}],"minor_comments":[{"comment":"In the condition on N, the expression 'min(1, /qomega,q)' is missing the parameter p and should read 'min(1, p/qomega, q)' or an analogous expression; as printed the inequality is meaningless.","section":"Theorem 4.6"},{"comment":"There are several typographical errors that should be corrected: 'calss' for 'class', 'sapces' for 'spaces', 'Beov' for 'Besov', 'charaterizations' for 'characterizations', 'Theorme' for 'Theorem', and 'Dnu' for 'D_nu' in the proof of Theorem 4.3.","section":"Throughout"},{"comment":"The phrase 'the set of all sequences all sequences {g_j}' contains a duplicated phrase and should read 'the set of all sequences {g_j}'.","section":"Definition 3.5"},{"comment":"The sentence 'Theorems 3.8 says that...' should be 'Theorem 3.8 says that...'.","section":"Remark 3.5"},{"comment":"The exponent gamma is stated using qomega, but since omega in A_infty may not lie in A_{qomega}, the statement should either use an arbitrary u > qomega as in Theorem 3.9 and Corollary 3.2, or justify why qomega itself is admissible.","section":"Theorem 3.10"},{"comment":"In the definition of a_Q, the formula divides by s_Q; the case s_Q = 0 should be excluded or handled by a standard convention, since otherwise the atom is not defined.","section":"Equation (4.2)"}],"recommendation":"major_revision","confidential_remarks":"The paper builds heavily on [20] and on [3] by two of the authors, but the dependence is transparent and not circular. The main concern is the incomplete proof of Theorem 3.9, which invalidates Theorem 3.10 for q != 2; this is fixable by either providing a correct argument for the full range or by restricting the stated characterizations to q=2. The dyadic-system independence issue also needs to be addressed. Given the scope of the claimed results, major revision seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Checked the paper and the stress-test note. The stress-test lands: Theorem 3.9 is not proved for q≠2.\n\nWhat's genuinely new: the type spaces with cube-truncated frequency sums (j ≥ j_Q) are not in [3] or [20]. The paper runs the full standard program—Peetre maximal functions, compact-support functional calculus, heat kernels, atomic/molecular decompositions, embeddings, completeness—and most of it is competent. The Hardy-type inequality (Theorem 3.3) is a clean adaptation of Yang-Yuan's lemma and is used well. The applications to fractional powers and spectral multipliers are straightforward but fit.\n\nThe gap: the proof of Theorem 3.9 reduces to s=0, q=2 by setting \\tilde F = (α^{-s}|F|)^{q/2}. That substitution does not preserve the quantities being compared. Applying the q=2 estimate to \\tilde F with exponent 2p/q yields a Bourgain-Morrey norm with integrability 2p/q, not p, and the outer exponent shifts; applying it with exponent p puts the wrong power inside. Raising the whole inequality to a power cannot repair the mismatch over the cube sum. So Theorem 3.9 is established only for q=2, and Corollary 3.2 and Theorem 3.10, which invoke it for all 0<q≤p, are unsupported for q≠2. This is an internal gap in the argument, not a parameter restriction.\n\nMinor issues: Theorem 4.6 has a typo in the condition on N (missing p in the denominator); the type spaces depend on a fixed dyadic system, and no independence proof is given; the borrowing from [20] and the authors' own [3] is heavy, acknowledged, and mostly legitimate, but it leaves the genuinely new parts concentrated in the definitions and the Hardy inequality.\n\nThe rest of the paper—Peetre and heat-kernel characterizations, atomic/molecular theory, completeness—does not rest on Theorem 3.9 and looks sound. So this is a fixable paper, not a reject. Send it to a serious referee; the referee should insist on a correct proof or a restricted statement for Theorem 3.9.","headline":"Useful technical extension with a real gap: Theorem 3.9's reduction to q=2 does not work, so the Lusin/Littlewood-Paley characterizations are unsupported for q≠2.","tokens_in":48920,"tokens_out":4824,"would_cite":false,"duration_ms":42949,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E36","46F05","47B38"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces weighted homogeneous Bourgain-Morrey-Besov and Triebel-Lizorkin type spaces associated with a nonnegative self-adjoint operator $L$ and proves that they admit Peetre maximal-function, heat-kernel, atomic, molecular…","keywords":["Bourgain-Morrey spaces","Besov-type spaces","Triebel-Lizorkin-type spaces","operators with Gaussian heat kernel","Peetre maximal functions","atomic decomposition","molecular decomposition","Muckenhoupt weights"],"falsifier":"Take $X=\\mathbb{R}^n$ with $L=-\\Delta$, choose $p<t<r<\\infty$ in the allowed range and a Muckenhoupt weight $\\omega\\in A_p$, and compute both sides of the norm equivalence in Theorem 3.5 for a Schwartz test function whose Littlewood-Paley pieces are known explicitly. If the Peetre maximal-function norm and the discrete frequency norm fail to be comparable for some such triple, or if the ratio blows up as $r$ approaches $-nt/\\log\\beta$, the central characterization is false.","tokens_in":47848,"feed_emoji":"🧮","tokens_out":8579,"duration_ms":67517,"temperature":0.7,"pith_summary":"This paper aims to extend the weighted theory of Besov and Triebel-Lizorkin spaces attached to a nonnegative self-adjoint operator $L$ to Bourgain-Morrey-type function spaces. On a space of homogeneous type with infinite measure, doubling, and a reverse doubling condition, and under a Gaussian upper bound for the heat kernel of $L$, the authors introduce dyadic-cube norms whose local $L^p(\\omega)$ pieces are measured against the weight $\\omega(Q)^{1/t-1/p}$ and summed over all scales and frequencies. They prove that the resulting spaces are independent of the Schwartz function used in the Littlewood-Paley decomposition and can be characterized by Peetre maximal functions, by noncompactly supported functional calculus, and by heat-kernel means. They also establish atomic and molecular decompositions and, as applications, show that fractional powers of $L$ shift the smoothness index while spectral multipliers of Laplace-transform type act boundedly. A sympathetic reader would care because this is a step toward using these rougher, Morrey-like spaces in PDE settings where classical Besov spaces are too fine.","feed_headline":"Operator-linked Besov-Morrey spaces get atomic decompositions","feed_subtitle":"New weighted spaces tied to $L$ now have Peetre, heat-kernel, and spectral-multiplier characterizations.","key_machinery":"The load-bearing mechanism is the sequence-valued weighted Bourgain-Morrey norm $\\hat\\ell^q(M^{t,r}_{p,\\omega})$ and its Triebel-Lizorkin counterpart $\\widehat{M}^{t,r}_{p,\\omega}(\\ell^q)$: for each dyadic cube $Q$ with scale $j_Q$, the frequency sum starts only at $j=j_Q$, and the cube weight $\\omega(Q)^{1/t-1/p}$ converts local Lebesgue norms into Morrey-type information. The Hardy type inequality (Theorem 3.3) and the maximal-function bounds (Theorems 3.1 and 3.2), valid under the reverse-doubling parameter restriction $r > -nt/\\log\\beta$, let the authors compare different Littlewood-Paley decompositions via Peetre maximal functions (Theorem 3.5). The heat-kernel estimates then reduce to the Schwartz-class functional calculus, and the atomic and molecular decompositions are built from dyadic cubes together with the kernel estimates of [20].","core_discovery":"The central discovery is that the family of weighted homogeneous Bourgain-Morrey-Besov type spaces $\\dot B^{s,q,L}_{p,t,r,\\omega}(X)$ and Triebel-Lizorkin type spaces $\\dot F^{s,q,L}_{p,t,r,\\omega}(X)$ are well-behaved function spaces for which the standard real-variable toolkit works. Theorem 3.5 gives norm equivalences with Peetre maximal functions; Theorems 3.7 and 3.8 give continuous characterizations via heat-kernel and noncompact-support functional calculus; Theorem 3.10 gives Lusin and Littlewood-Paley characterizations for the Triebel-Lizorkin type spaces; Theorems 4.1 through 4.6 give atomic and molecular decompositions; and Theorems 5.1 and 5.2 give boundedness of fractional powers and spectral multipliers. Together these results show that the spaces are complete, continuously embedded into the distribution space $\\mathcal S'_\\infty$ modulo polynomials, and stable under the natural functional calculus of $L$.","pith_inferences":["The parameter restriction $r > -nt/\\log\\beta$ suggests a critical integrability window tied to the weight and the reverse-doubling constant; outside this window even maximal-function boundedness may fail, so the theory likely cannot be extended without replacing Muckenhoupt weights by a more restrictive class or adding a different geometric condition.","Because Bourgain-Morrey spaces arise in Strichartz estimates and nonlinear Schr\\\"odinger equations, the proven boundedness of fractional powers and spectral multipliers on these spaces may support well-posedness and regularity arguments for dispersive PDE, although the paper does not pursue such applications.","The dyadic-cube-based definitions depend on a fixed cube system; proving independence from the choice of cubes beyond the adjacent systems already handled in Lemma 3.3 would be needed before fully coordinate-free applications, and the existing Lemma 3.3 suggests this may be within reach.","In the limiting case $p=t$ and $r=\\infty$, the new spaces reduce to the weighted Besov and Triebel-Lizorkin spaces of [20], so the paper can be read as a Morrey-type interpolation between those spaces and Bourgain-Morrey spaces; explicit interpolation or embedding results between these families are a natural next step."],"forward_implications":["The spaces are complete and continuously embedded into $\\mathcal S'_\\infty$, so they can serve as distribution spaces for PDE on metric measure spaces.","Fractional powers $L^{\\tau/2}$ map $\\dot B^{s,q,L}_{p,t,r,\\omega}(X)$ continuously into $\\dot B^{s+\\tau,q,L}_{p,t,r,\\omega}(X)$, and similarly for the Triebel-Lizorkin type spaces.","Spectral multipliers of Laplace-transform type are bounded on both families of spaces.","Atomic and molecular decompositions imply that the test space $\\mathcal S_\\infty$ is dense in the new spaces, which is useful for approximation and for transferring results from smooth functions."],"supporting_citations":[{"why":"Supplies the template for weighted Besov and Triebel-Lizorkin spaces associated with $L$, including Peetre maximal-function and heat-kernel characterizations, atomic and molecular decompositions, and the fractional-power and spectral-multiplier arguments adapted here.","marker":"[20]"},{"why":"Establishes the weighted Bourgain-Morrey spaces, their maximal-function boundedness, and the predecessor Bourgain-Morrey-Besov and Triebel-Lizorkin spaces whose definitions and norms are used in Section 3.","marker":"[3]"},{"why":"Provides the dyadic cube decomposition on spaces of homogeneous type that gives the fixed cube system $\\mathcal D$ used throughout the paper.","marker":"[27]"},{"why":"Supplies adjacent dyadic systems and the comparison of the Hardy-Littlewood maximal function with dyadic maximal functions, used in the proof of Theorem 3.1.","marker":"[33]"},{"why":"Origin of the Hardy type inequality argument for sequence spaces, adapted here as Theorem 3.3 to control the weighted Bourgain-Morrey norms.","marker":"[83]"},{"why":"Provides the integral estimates for $V(x,\\alpha)$ and the maximal-function bound used to pass from Peetre maximal functions to Hardy-Littlewood maximal functions.","marker":"[25]"},{"why":"Sets up the test-function and distribution spaces $\\mathcal S$, $\\mathcal S_\\infty$, and $\\mathcal S'_\\infty$ and the identification modulo polynomials needed to define the operator functional calculus.","marker":"[39]"},{"why":"Gives the heat-kernel based decomposition and functional calculus for self-adjoint operators, used in representing distributions and in kernel estimates.","marker":"[50]"}],"fun_headline_variants":["Weighted operator spaces get atomic and molecular decompositions","Operator-tied Bourgain-Morrey spaces yield atomic decompositions","New weighted Besov-Morrey spaces get continuous characterizations","Atomic decompositions for operator-based Triebel-Lizorkin spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the reverse doubling condition for $(X,\\mu)$ and on the parameter restriction $r > -nt/\\log\\beta$, where $\\beta$ is a self-improvement constant of the Muckenhoupt weight; without these, the maximal-function estimates and hence the norm equivalences are not proved.","fun_headline_variants_meta":{"raw":{"variants":["Weighted operator spaces get atomic and molecular decompositions","Operator-tied Bourgain-Morrey spaces yield atomic decompositions","New weighted Besov-Morrey spaces get continuous characterizations","Atomic decompositions for operator-based Triebel-Lizorkin spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00033,"raw_usage":{"total_tokens":1837,"prompt_tokens":942,"completion_tokens":895,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":824}},"tokens_in":558,"tokens_out":895,"duration_ms":11163,"temperature":1.0,"reasoning_tokens":824,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:19:25.930187+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $X=\\mathbb{R}^n$ with $L=-\\Delta$, choose $p<t<r<\\infty$ in the allowed range and a Muckenhoupt weight $\\omega\\in A_p$, and compute both sides of the norm equivalence in Theorem 3.5 for a Schwartz test function whose Littlewood-Paley pieces are known explicitly. If the Peetre maximal-function norm and the discrete frequency norm fail to be comparable for some such triple, or if the ratio blows up as $r$ approaches $-nt/\\log\\beta$, the central characterization is false.","supporting_citations":[{"cited_title":"Weighted Besov and Triebel-Lizorkin spaces asso- ciated with operators and applications","cited_arxiv_id":null,"evidence_quote":"Supplies the template for weighted Besov and Triebel-Lizorkin spaces associated with $L$, including Peetre maximal-function and heat-kernel characterizations, atomic and molecular decompositions, and the fractional-power and spectral-multiplier arguments adapted here."},{"cited_title":"A T (b) theorem with remarks on analytic capacity and the Cauchy inte- gral","cited_arxiv_id":null,"evidence_quote":"Provides the dyadic cube decomposition on spaces of homogeneous type that gives the fixed cube system $\\mathcal D$ used throughout the paper."},{"cited_title":"The boundedness of fractional maximal operators on vari- able Lebesgue spaces over spaces of homogeneous type","cited_arxiv_id":null,"evidence_quote":"Supplies adjacent dyadic systems and the comparison of the Hardy-Littlewood maximal function with dyadic maximal functions, used in the proof of Theorem 3.1."},{"cited_title":"Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means","cited_arxiv_id":null,"evidence_quote":"Origin of the Hardy type inequality argument for sequence spaces, adapted here as Theorem 3.3 to control the weighted Bourgain-Morrey norms."},{"cited_title":"Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type","cited_arxiv_id":null,"evidence_quote":"Provides the integral estimates for $V(x,\\alpha)$ and the maximal-function bound used to pass from Peetre maximal functions to Hardy-Littlewood maximal functions."},{"cited_title":"Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators","cited_arxiv_id":null,"evidence_quote":"Sets up the test-function and distribution spaces $\\mathcal S$, $\\mathcal S_\\infty$, and $\\mathcal S'_\\infty$ and the identification modulo polynomials needed to define the operator functional calculus."},{"cited_title":"Heat kernel based decomposition of spaces of dis- tributions in the framework of Dirichlet spaces","cited_arxiv_id":null,"evidence_quote":"Gives the heat-kernel based decomposition and functional calculus for self-adjoint operators, used in representing distributions and in kernel estimates."}],"review_version":1}