{"paper":{"title":"The nonlinear estimates on quantum Besov spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Superposition operators with non-smooth symbols are bounded on quantum Besov spaces.","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Deyu Chen, Guixiang Hong","submitted_at":"2026-01-17T06:48:17Z","abstract_excerpt":"The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results \\cite{McNLE} for infinitely differentiable symbols and have rich applications in the well-posedness theory of noncommutative PDEs. The ingredients in the proof involve a novel quantum chain rule and nonlinear interpolation. As a byproduct, we prove the equivalence of "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces... As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in [Remark 3.16]{McNLE}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The non-smooth symbols still satisfy the minimal regularity needed for the novel quantum chain rule and nonlinear interpolation to apply without additional restrictions that would limit the claimed generality.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Boundedness of superposition operators with non-smooth symbols is established on quantum Besov spaces, together with equivalence of two space descriptions that resolves a prior conjecture.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Superposition operators with non-smooth symbols are bounded on quantum Besov spaces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d4e4591f19b0309b21cdda852d977fafd334d73920e9cbed4f2c524b11e17dd8"},"source":{"id":"2601.11934","kind":"arxiv","version":2},"verdict":{"id":"fad5c87f-5e7a-4c6b-9ac6-7f6046feed3b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T13:28:49.450834Z","strongest_claim":"We investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces... As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in [Remark 3.16]{McNLE}.","one_line_summary":"Boundedness of superposition operators with non-smooth symbols is established on quantum Besov spaces, together with equivalence of two space descriptions that resolves a prior conjecture.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The non-smooth symbols still satisfy the minimal regularity needed for the novel quantum chain rule and nonlinear interpolation to apply without additional restrictions that would limit the claimed generality.","pith_extraction_headline":"Superposition operators with non-smooth symbols are bounded on quantum Besov spaces."},"references":{"count":78,"sample":[{"doi":"","year":2011,"title":"W. Arendt, C. Batty, M. Hieber and F. Neubrander. Vector-Valued Laplace Transforms and Cauchy Problems. Second edition, Monogr. Math., V ol. 96, Birkh¨auser/Springer, Basel AG, Basel, 2011, MR2798103.","work_id":"8a9914c9-69df-461b-9962-c8aae9531edd","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"J. Appell and P. P. Zabrejko. Nonlinear superposition operators. V ol. 95 ofCambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1990. 3","work_id":"6a8ac7de-e751-442b-99ec-1b7729d732d7","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"N. A. Azamov, A. L. Carey, P. G. Dodds, and F. A. Sukochev. Operator Integrals, Spectral Shift, and Spectral Flow.Canad. J. Math.,61(2009), no.2, 241–263. 6, 20, 21, 23, 25, 27","work_id":"e386fd59-d44d-4992-aa1a-96f9823f023b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"S. Bartels. Numerical Methods for Nonlinear Partial Differential Equations. Deutschland: Springer International Publishing, (2015). 3","work_id":"e062df4b-11f1-4a9c-8af1-85629dcf9e65","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"K. Bamba, M. Saitou and A. Sugamoto. Hydrodynamics on non commutative space: A step toward hydrodynam- ics of granular materials.Prog. Theor. Exp. Phys.,10(2014), 103B03. 2 THE NONLINEAR ESTIMATES ON ","work_id":"8955bf7f-ab86-4c22-ba6c-d2cb9d3bf5cf","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":78,"snapshot_sha256":"06dfd70f13a2e6b153fb8946799870ee07db21fcdd9f7bad9ab5e7d6855dac4e","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"5dc4b88ae181c9c405a98a1cd6844778696c8ca6b8b10a4b02e6c2a2ddf43b8e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}