{"id":"6349e524-e9b5-414c-b2b2-0f0ac76262d4","arxiv_id":"2506.05493","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"New orthonormal Strichartz bounds for the Dunkl-Schrödinger propagator are claimed for initial data in homogeneous Dunkl-Sobolev spaces.","lead":"This paper proves orthonormal Strichartz estimates for the Schrödinger equation governed by the Dunkl Laplacian, allowing initial data with Sobolev regularity rather than plain square-integrable data. If correct, the results extend known Euclidean estimates to weighted reflection-symmetric spaces, which are relevant to many-particle quantum systems and harmonic analysis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.5 is false: the claimed Young-type inequality for the kernel ||x|-|y|| fails even in the Euclidean case κ=0, n=2, and the non-radiality of hκ is only one symptom; Lemma 5.2 and Theorems 1.4–1.5 rest on it.","rationale":"In good faith, the central claim is an orthonormal Strichartz estimate in Dunkl-Sobolev spaces, and the proof strategy is plausible: use Lorentz-space refinements, frequency localization, and interpolation. The reader identified Lemma 3.5 as the load-bearing assumption, specifically the radiality of hκ. I agree that Lemma 3.5 is the critical point, but the concern is stronger than radiality. Even with a radial weight — indeed even with hκ=1 — the inequality stated in Lemma 3.5 is not a consequence of Young's inequality, because the kernel f~(||x|-|y||) depends on the difference of radial coordinates rather than on |x-y|. The explicit two-dimensional approximate-identity computation shows the claimed L^r bound fails for p=q=3/2, r=3, which is exactly the regime used in Lemma 5.2. This is an internal inconsistency, not a disagreement with consensus: for κ=0 the final theorem is known, but the submitted derivation does not establish it. The proof of Lemma 5.2's T_j^{(0)} and T_j^{(1)} estimates, and hence Theorem 1.4 and Theorem 1.5, relies on the false lemma. I therefore see no reason to change the reader's REJECT verdict; the concern confirms it rather than moving it elsewhere.","tokens_in":30832,"tokens_out":16182,"duration_ms":141920,"concrete_test":"Run the stated counterexample for Lemma 3.5: set κ=0, n=2, p=q=3/2, r=3, fε(x)=π^{-2/3}ε^{-4/3}1_{|x|≤ε}, gε(y)=(2πRε)^{-2/3}1_{R≤|y|≤R+ε}, and compute Gε(x)=∫fε(||x|-|y||)gε(y)dy. If the numerically or analytically evaluated norm satisfies ∥Gε∥_{L^3(R^2)} ≳ R^{2/3}ε^{-2/3}, the lemma is refuted and the frequency-localized estimates in Lemma 5.2 lack a valid proof. A positive result would require replacing ||x|-|y|| by the Euclidean distance |x-y| or adding hypotheses on the weight and kernel such as those of the classical radial Young inequality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's global estimates depend on the frequency-localized bounds in Lemma 5.2, whose proof invokes Lemma 3.5 for the kernel f~(||x|-|y||). Lemma 3.5 is the weakest link, and it is false in two independent ways. First, hκ is not radial for n≥2 and any nontrivial Coxeter root system; for type A1 in R^2, hκ(x)=|x1|^{2κ}, so the proof's reduction to radial coordinates is invalid. Second, and more seriously, the asserted inequality fails even when hκ=1 (κ=0) and g is radial. In R^2 with p=q=3/2, r=3, take fε(x)=π^{-2/3}ε^{-4/3}1_{|x|≤ε} and gε(y)=(2πRε)^{-2/3}1_{R≤|y|≤R+ε}, both L^{3/2}-normalized. Then Gε(x)=∫fε(||x|-|y||)gε(y)dy is supported near the annulus |x|≈R and satisfies ∥Gε∥_{L^3(R^2)} ≳ R^{2/3}ε^{-2/3}, which contradicts the bound ∥Gε∥_{L^3}≤∥fε∥_{L^{3/2}}∥gε∥_{L^{3/2}}=1 as R≫ε^{-1}. Thus Lemma 3.5 is not a harmless radiality slip; it is a false convolution inequality for Euclidean distance |x-y| incorrectly replaced by the radial-coordinate difference ||x|-|y||. Since Lemma 5.2's bounds for T_j^{(0)} and T_j^{(1)} use this lemma directly, the proofs of Theorem 1.4 and Theorem 1.5 are unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops frequency-localized and global orthonormal Strichartz estimates for the Dunkl-Schrödinger propagator with initial data in homogeneous Dunkl-Sobolev spaces, extending results of Frank–Sabin and Bez–Hong–Lee–Nakamura–Sawano to the Dunkl setting. The main results are Theorem 1.4 (frequency-localized estimates for N>2, with remarks for 1≤N≤2) and Theorem 1.5 (global strong-type estimates). The argument combines Lorentz-space refinements of known Strichartz estimates, Schatten-space duality, real and complex interpolation, and a new inequality stated as Lemma 3.5. The paper is organized around these ingredients and includes the relevant preliminaries on Dunkl analysis.","tokens_in":31269,"tokens_out":6220,"duration_ms":64685,"significance":"The target results, if true, would be a meaningful refinement of existing orthonormal Strichartz estimates in the Dunkl setting: they allow Sobolev-regular initial data with the correct scaling relation 2s=N-(2/q+N/p) and cover the region int OAB and int OCDA. The paper contains no fitted parameters and does not assume the desired inequality as an input; it builds on known dispersive estimates and interpolation machinery. The frequency-localized strategy is coherent, and the Lorentz-space refinements in Theorem 4.1 and Theorem 4.2 are natural intermediate steps. However, the central technical lemma on which the frequency-localized argument rests is false, and consequently the claimed main results are not established.","major_comments":[{"comment":"Lemma 3.5 is the load-bearing ingredient in the proof of Lemma 5.2, where it is used to bound both T_j^{(0)} and T_j^{(1)}, and its proof is invalid because it asserts that h_κ is radial. The definition in §2.1, h_κ(x)=∏_{α∈R_+}|⟨α,x⟩|^{2κ_α}, is not radial for n≥2 and any nontrivial finite reflection group; for example, the type A_1 root system in R^2 gives h_κ(x)=|x_1|^{2κ}. Consequently the identity (3.9) with a single angular constant C_κ and the reduction of ∥G∥_{L^r_κ} to a one-dimensional weighted norm are unjustified. Since Lemma 5.2 is used directly in the proof of Theorem 1.4 and then in the proof of Theorem 1.5, the frequency-localized and global estimates are unsupported.","section":"Section 3, Lemma 3.5"},{"comment":"Independently of the radiality issue, the asserted inequality is false even in the Euclidean case κ=0. For n=2, p=q=3/2 and r=3, take f_ε(x)=π^{-2/3}ε^{-4/3}1_{|x|≤ε} and g_ε(y)=(2π Rε)^{-2/3}1_{R≤|y|≤R+ε}, both normalized in L^{3/2}. Then G_ε(x)=∫ f_ε(||x|-|y||)g_ε(y)dy is supported in an annulus of radius roughly R and thickness roughly ε, with pointwise size about R^{1/3}ε^{-1}; hence ∥G_ε∥_{L^3(R^2)} is comparable to R^{2/3}ε^{-2/3}, which contradicts the claimed bound by ∥f_ε∥_{L^{3/2}}∥g_ε∥_{L^{3/2}}=1 when R≫ε^{-1}. The same translation-invariance failure already appears in the proof's intermediate identification of the half-line convolution norm with the radial L^p_κ norm, since the weight ρ^{N-1}dρ is not translation-invariant. Lemma 3.5 is therefore not a harmless radiality slip, and the parts of Lemma 5.2 that invoke it must be replaced.","section":"Section 3, Lemma 3.5"}],"minor_comments":[{"comment":"The displayed relation '2s = 2((1−θ)·0+θs1)' should read '2s = θ(2s1)' or equivalently '2s = N−(2/q+N/p)'; as printed, the equality is dimensionally inconsistent.","section":"Section 6, proof of Theorem 6.2"},{"comment":"The heading 'Proof of Remark 1.1, 1.1 and 1.1' and the later use of 'Remark 1.1, 1.1 and 1.1' in the proof of Theorem 6.1 refer to the remarks appended to Theorem 1.4, not to Remark 1.1; these cross-references need correction.","section":"Section 6"},{"comment":"There is a duplicated word in the sentence 'In addition, if if (1/p,1/q) belongs to the region int OAF ...'.","section":"Section 1, after Notation 1.3"},{"comment":"The dyadic decomposition is written as 'P_{j∈R} T_j'; the index set should be Z, as used throughout the subsequent argument.","section":"Section 5, proof of Theorem 1.4"},{"comment":"In the line 'As the Dunkl-Schrödinger operator e^{itΔκ} is unitary on L^2(R^n)', the space should be L^2_κ(R^n) with the weighted measure.","section":"Section 2.4"}],"recommendation":"reject","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper aims to extend the Bez–Hong–Lee–Nakamura–Sawano orthonormal Strichartz estimates with Sobolev regularity to the Dunkl setting, and it adds Lorentz-space refinements. The goal is sensible, the interpolation architecture is standard, and the paper follows [2] and [14] closely. If the main theorem were true, it would be a useful advance for Dunkl harmonic analysis and for Hartree-type applications. But the central technical lemma, Lemma 3.5, is false, and the proof leans on it at every critical step.\n\nThe lemma claims a Young-type inequality for the kernel f~(|x| − |y|) with the weighted measure h_kappa(x) dx. Two things go wrong. First, h_kappa is not radial for n ≥ 2 with a nontrivial Coxeter group; for type A1 in R^2, h_kappa(x) = |x1|^{2kappa}, so the reduction to G(x) = G~(rho) in the proof is invalid. Second, the inequality itself fails even when kappa = 0 and h = 1. In R^2 with p = q = 3/2, r = 3, take f_epsilon normalized on |x| ≤ epsilon and g_epsilon on the annulus R ≤ |y| ≤ R + epsilon. The resulting G is roughly epsilon^{-1} R^{1/3} on an annulus of area O(R epsilon), so its L^3 norm is ∼ R^{2/3} epsilon^{-2/3}, violating the claimed bound as R ≫ epsilon^{-1}. So the lemma is not a harmless radiality slip; it is simply not a valid convolution inequality.\n\nThis is load-bearing. Lemma 5.2 invokes Lemma 3.5 directly for the bounds on T_j^{(0)} and T_j^{(1)}. Theorem 1.4's frequency-localized estimates rest on Lemma 5.2, and Theorem 1.5's global estimates rest on Theorem 1.4 via the interpolation machinery. Once Lemma 3.5 is removed, the proof has no route from the L^2_kappa-based estimates of [29] to the Sobolev-regularity claims.\n\nThe paper is not sloppy about the literature: it correctly identifies [2] for Euclidean Sobolev-regularity orthonormal Strichartz and [29] for L^2_kappa Dunkl results, and there is no circularity or parameter fitting. But the main results are unsupported as written.\n\nWho is this for? Only someone working on Dunkl harmonic analysis or orthonormal Strichartz variants, and even then as a cautionary example. It should not be published without a genuine repair of Lemma 3.5. If the authors can replace it with a valid radial-difference inequality, the rest of the framework might survive; as it stands, a serious referee should not pass it. I would not cite it.","headline":"The main estimates are unsupported: Lemma 3.5 is a false convolution inequality, and the frequency-localized proof collapses on it.","tokens_in":31756,"tokens_out":4418,"would_cite":false,"duration_ms":51345,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E25","33C45","35H20","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that Dunkl-Schrödinger orthonormal Strichartz estimates extend from $L^2_\\kappa$ data to homogeneous Sobolev data, with the sequence exponent $\\alpha$ set by scaling.","keywords":["Dunkl Laplacian","orthonormal Strichartz estimates","homogeneous Dunkl-Sobolev space","Lorentz spaces","Schatten classes","frequency-localized estimates","real and complex interpolation","Schrödinger equation"],"falsifier":"A decisive check is Lemma 3.5 in a case it cannot handle: on $\\mathbb R^2$ with root system $\\{\\pm e_1\\}$ and weight $h_\\kappa(y)=|y_1|^{2k}$, take $\\tilde f$ a smooth bump near radius 1 and $g(y)=|y_1|$, a non-radial $L^q$ function, and compare $G((a,0))$ with $G((0,a))$ for $a>0$. If these values differ, $G$ is not radial and the lemma's proof fails; because Lemma 5.2 uses Lemma 3.5 to bound $T_j^{(0)}$ and $T_j^{(1)}$, the frequency-localized proof of Theorem 1.4 is then not valid as written, and a numerical test of the target estimate at the exponents predicted by Theorem 1.5 would show whether the theorem itself survives.","tokens_in":30676,"feed_emoji":"🧮","tokens_out":15714,"duration_ms":163213,"temperature":0.7,"pith_summary":"The paper claims that the Dunkl–Schrödinger propagator $e^{it\\Delta_\\kappa}$ obeys orthonormal Strichartz estimates for initial data in the homogeneous Dunkl–Sobolev space $\\dot H^s_\\kappa(\\mathbb R^n)$, not only for $L^2_\\kappa$ data. For exponent pairs $(1/p,1/q)$ in the interior of the triangle $OAB$ and for the Sobolev order fixed by the scaling relation $2s=N-(2/q+N/p)$, it asserts the bound $$\\Big\\|\\sum_j \\lambda_j |$e^{{it\\Delta_\\kappa}}$ f_j|^2\\Big\\|_{L^q_t(\\mathbb R,L^p_\\kappa(\\mathbb R^n))} \\lesssim \\|\\{\\lambda_j\\}\\|_{\\ell^\\$\\alpha$}$$ with $\\alpha=\\alpha^*(p,q)$ determined by $N/\\alpha=1/q+N/p$, for every orthonormal family $\\{f_j\\}$ in $\\dot H^s_\\kappa$. A companion statement covers the region $\\operatorname{int} OCDA$ with $\\alpha<q$. The significance is that this would extend the Euclidean Sobolev-regularity orthonormal Strichartz estimates for the classical Laplacian to the weighted Dunkl setting, and it reduces to the earlier $L^2_\\kappa$ result when $s=0$.","feed_headline":"Orthonormal Dunkl-Schrödinger bounds reach Sobolev data","feed_subtitle":"Orthogonal systems of initial states in a weighted Sobolev space satisfy spacetime bounds with the scaling-optimal ℓ^α exponent.","key_machinery":"The carrying object is the Dunkl–Laplacian $\\Delta_\\kappa$, the reflection-invariant differential-difference operator obtained by replacing coordinate derivatives with Dunkl derivatives, together with its Schrödinger propagator $e^{it\\Delta_\\kappa}$, a unitary group on the weighted space $L^2_\\kappa(\\mathbb R^n)$ with weight $h_\\kappa(x)=\\prod_{\\alpha\\in R_+}|\\langle\\alpha,x\\rangle|^{2k_\\alpha}$. The argument moves through four main mechanisms: a duality principle converting orthonormal Strichartz estimates into Schatten-class bounds for the operator $W e^{it\\Delta_\\kappa}(e^{it\\Delta_\\kappa})^* W$; a refined Young-type inequality (Lemma 3.5) for integrals with kernel $\\tilde f(||x|-|y||)$; dyadic Littlewood–Paley projections $P_k$ and their frequency-localized estimates (Theorem 1.4); and successive real and complex interpolation, including bilinear interpolation, that assembles the localized bounds into restricted weak-type and then strong-type global estimates (Theorem 1.5).","core_discovery":"On the paper's own terms, the central discovery is that the Sobolev restriction $2s=N-(2/q+N/p)$ is compatible with systems of orthonormal Dunkl–Schrödinger waves: the estimate $$\\Big\\|\\sum_j \\lambda_j |$e^{{it\\Delta_\\kappa}}$ f_j|^2\\Big\\|_{L^q_t(\\mathbb R,L^p_\\kappa(\\mathbb R^n))} \\lesssim \\|\\{\\lambda_j\\}\\|_{\\ell^\\$\\alpha$}$$ holds for $(1/p,1/q)\\in \\operatorname{int} OAB$ with $\\alpha=\\alpha^*(p,q)$, and the analogous statement holds in $\\operatorname{int} OCDA$ with $\\alpha<q$. The proof first establishes frequency-localized estimates for data supported on dyadic annuli, using dispersive decay of the propagator together with a refined Young-type inequality for radial kernels, and then upgrades to global Sobolev data through restricted weak-type interpolation and a chain of real and complex interpolation steps. When $s=0$ the exponent reduces to $\\alpha=2p/(p+1)$, matching the known $L^2_\\kappa$ orthonormal Strichartz bound; when $\\kappa=0$ the Dunkl operator coincides with the Euclidean Laplacian and the statements reduce to the classical Sobolev-regularity orthonormal Strichartz estimates.","pith_inferences":["If Theorem 1.5 holds, the same interpolation chain should work for any self-adjoint operator whose propagator has the same dispersive decay; the only place the proof uses the specific Dunkl structure is the radial kernel inequality, so replacing that lemma would carry the method to other weighted settings.","A plausible endpoint question the paper leaves open is whether the estimate persists on the boundary of $OAB$, where the classical Euclidean cases often require separate arguments.","If the Sobolev-regularity estimates are valid, they should give Schatten-space well-posedness for Dunkl analogues of Hartree equations with initial data below $L^2_\\kappa$ regularity, following the usual fermionic many-body route.","The scaling formula $N/\\alpha=1/q+N/p$ pins the sequence exponent, so any strengthening of Theorem 1.5 would have to exploit structure beyond scaling."],"forward_implications":["When $\\kappa=0$ the Dunkl–Laplacian is the Euclidean Laplacian, so Theorem 1.5 reduces to the orthonormal Sobolev-space Strichartz estimates known for the classical Schrödinger equation.","Taking only one nonzero coefficient in the orthonormal family recovers the single-function Dunkl–Strichartz estimate in homogeneous Sobolev spaces, including fractional regularity $s$.","The exponent $\\alpha=\\alpha^*(p,q)$ is the one forced by scaling, so the $\\ell^\\alpha$ dependence is sharp whenever the estimates hold.","The frequency-localized estimates of Theorem 1.4 give annulus-localized orthonormal bounds that, through the duality principle, imply Schatten-space bounds for the Dunkl–Schrödinger propagator.","The Lorentz-space refinements proved along the way are stronger than the Lebesgue-space statements and are needed to make the interpolation chain work."],"supporting_citations":[{"why":"Establishes the classical orthonormal Sobolev Strichartz estimates for the Euclidean Laplacian that this paper's global result extends to the Dunkl setting.","marker":"[2]"},{"why":"Introduces the orthonormal Strichartz inequality for systems of orthogonal functions in $L^2$ data, the baseline being generalized here.","marker":"[13]"},{"why":"Supplies the duality principle converting orthonormal estimates into Schatten-class bounds and the complex interpolation machinery for Schatten spaces.","marker":"[14]"},{"why":"Provides the endpoint Strichartz estimates and the bilinear real-interpolation model used on the critical frequency-localized segment.","marker":"[25]"},{"why":"Proves the single-function Dunkl-Strichartz estimates that the present paper extends to Sobolev regularity and to orthonormal systems.","marker":"[28]"},{"why":"Proves the $L^2_\\kappa$ orthonormal Dunkl-Strichartz estimates that Theorem 1.5 refines to homogeneous Sobolev data.","marker":"[29]"},{"why":"Gives the Hardy–Littlewood–Sobolev inequality in Lorentz spaces used to sharpen the Dunkl-Strichartz estimates.","marker":"[31]"},{"why":"Supplies the positive intertwining representation and the radial Dunkl translation formula used in the kernel estimates.","marker":"[34]"},{"why":"Provides the Dunkl convolution Young inequality and the $L^p$ boundedness of radial Dunkl translations used in the dispersive estimates.","marker":"[40]"}],"fun_headline_variants":["Sobolev initial data now satisfy orthonormal Dunkl-Schrödinger bounds","Orthonormal Strichartz proven for Dunkl-Schrödinger with Sobolev data","Dunkl-Schrödinger orthonormal estimates reach Sobolev-regular data","Orthonormal Dunkl bounds extend to Sobolev-regular initial states","Optimal α for orthonormal Dunkl-Schrödinger with Sobolev data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 3.5's assertion that the Dunkl weight $h_\\kappa(x)=\\prod_{\\alpha\\in R_+}|\\langle\\alpha,x\\rangle|^{2k_\\alpha}$ is radial, which makes $G(x)=\\int \\tilde f(||x|-|y||)g(y)h_\\kappa(y)dy$ radial and reduces its $L^r$ norm to a one-dimensional weighted norm; for $n\\ge 2$ and any nontrivial reflection group this is false, already for $h_\\kappa(x)=|x_1|^{2k}$ on $\\mathbb R^2$, and Lemma 3.5 feeds the frequency-localized bounds in Lemma 5.2 that support Theorem 1.5.","fun_headline_variants_meta":{"raw":{"variants":["Sobolev initial data now satisfy orthonormal Dunkl-Schrödinger bounds","Orthonormal Strichartz proven for Dunkl-Schrödinger with Sobolev data","Dunkl-Schrödinger orthonormal estimates reach Sobolev-regular data","Orthonormal Dunkl bounds extend to Sobolev-regular initial states","Optimal α for orthonormal Dunkl-Schrödinger with Sobolev data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002129,"raw_usage":{"total_tokens":8260,"prompt_tokens":938,"completion_tokens":7322,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":7208}},"tokens_in":554,"tokens_out":7322,"duration_ms":60377,"temperature":1.0,"reasoning_tokens":7208,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:21:18.673610+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is Lemma 3.5 in a case it cannot handle: on $\\mathbb R^2$ with root system $\\{\\pm e_1\\}$ and weight $h_\\kappa(y)=|y_1|^{2k}$, take $\\tilde f$ a smooth bump near radius 1 and $g(y)=|y_1|$, a non-radial $L^q$ function, and compare $G((a,0))$ with $G((0,a))$ for $a>0$. If these values differ, $G$ is not radial and the lemma's proof fails; because Lemma 5.2 uses Lemma 3.5 to bound $T_j^{(0)}$ and $T_j^{(1)}$, the frequency-localized proof of Theorem 1.4 is then not valid as written, and a numerical test of the target estimate at the exponents predicted by Theorem 1.5 would show whether the theorem itself survives.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the classical orthonormal Sobolev Strichartz estimates for the Euclidean Laplacian that this paper's global result extends to the Dunkl setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the orthonormal Strichartz inequality for systems of orthogonal functions in $L^2$ data, the baseline being generalized here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the duality principle converting orthonormal estimates into Schatten-class bounds and the complex interpolation machinery for Schatten spaces."},{"cited_title":"Keel and T","cited_arxiv_id":null,"evidence_quote":"Provides the endpoint Strichartz estimates and the bilinear real-interpolation model used on the critical frequency-localized segment."},{"cited_title":"Mejjaoli, Dispersion phenomena in Dunkl-Schr¨ odinger equation and applications, Serdica Math","cited_arxiv_id":null,"evidence_quote":"Proves the single-function Dunkl-Strichartz estimates that the present paper extends to Sobolev regularity and to orthonormal systems."},{"cited_title":"O’Neil, Convolution operators and L(p, q) spaces, Duke Math","cited_arxiv_id":null,"evidence_quote":"Gives the Hardy–Littlewood–Sobolev inequality in Lorentz spaces used to sharpen the Dunkl-Strichartz estimates."},{"cited_title":"R¨ osler,Positivity of Dunkl’s intertwining operator , Duke Math","cited_arxiv_id":null,"evidence_quote":"Supplies the positive intertwining representation and the radial Dunkl translation formula used in the kernel estimates."},{"cited_title":"Thangavelu and Y","cited_arxiv_id":null,"evidence_quote":"Provides the Dunkl convolution Young inequality and the $L^p$ boundedness of radial Dunkl translations used in the dispersive estimates."}],"review_version":1}