{"paper":{"title":"On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The nonlinear Wigner concentration problem admits an optimizer for any finite-measure phase space set and every p less than infinity.","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Erling A. T. Svela, Federico Stra, S. Ivan Trapasso","submitted_at":"2025-10-21T14:40:34Z","abstract_excerpt":"We prove that, for any measurable phase space subset $\\Omega\\subset\\mathbb{R}^{2d}$ with $0<|\\Omega|<\\infty$ and any $1\\le p < \\infty$, the nonlinear concentration problem $$ \\sup_{f \\in L^2(\\mathbb{R}^d)\\setminus\\{0\\}}\\frac{\\|Wf\\|_{L^p(\\Omega)}}{\\|f\\|_{L^2}^2}$$ admits an optimizer, where $Wf$ is the Wigner distribution of $f$. The main obstruction is that $Wf$ is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any measurable Ω ⊂ ℝ^{2d} with 0 < |Ω| < ∞ and any 1 ≤ p < ∞, the nonlinear concentration problem sup (||Wf||_{L^p(Ω)} / ||f||_{L^2}^2) admits an optimizer.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The new asymptotic formula that quantifies the limiting contribution to concentration over Ω from asymptotically separated wave packets holds and can be combined with concentration compactness for Heisenberg-type dislocations to restore the necessary upper semicontinuity (abstract, section on main proof strategy).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Existence of optimizers is established for the Wigner-distribution concentration functional over finite-measure phase-space sets for 1 ≤ p < ∞, with sharp constant 2^d attained at p = ∞.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The nonlinear Wigner concentration problem admits an optimizer for any finite-measure phase space set and every p less than infinity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4124ad858f6d5abc30d50928e15da46757e1fb0ae3bd93c22ae029f3c426f4f0"},"source":{"id":"2510.18683","kind":"arxiv","version":3},"verdict":{"id":"3441beb2-be73-4699-ab57-58fd1a7fc119","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T05:00:48.900187Z","strongest_claim":"For any measurable Ω ⊂ ℝ^{2d} with 0 < |Ω| < ∞ and any 1 ≤ p < ∞, the nonlinear concentration problem sup (||Wf||_{L^p(Ω)} / ||f||_{L^2}^2) admits an optimizer.","one_line_summary":"Existence of optimizers is established for the Wigner-distribution concentration functional over finite-measure phase-space sets for 1 ≤ p < ∞, with sharp constant 2^d attained at p = ∞.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The new asymptotic formula that quantifies the limiting contribution to concentration over Ω from asymptotically separated wave packets holds and can be combined with concentration compactness for Heisenberg-type dislocations to restore the necessary upper semicontinuity (abstract, section on main proof strategy).","pith_extraction_headline":"The nonlinear Wigner concentration problem admits an optimizer for any finite-measure phase space set and every p less than infinity."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.18683/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":32,"sample":[{"doi":"","year":2010,"title":"P. Boggiatto, G. De Donno, and A. Oliaro. Time-frequency representations of Wigner type and pseudo-differential operators.Trans. Amer. Math. Soc., 362(9), 2010","work_id":"dfd296cb-4486-47ed-9298-a7a000ea1ed7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"A. J. Bracken, H.-D. Doebner, and J. G. Wood. Bounds on integrals of the wigner function.Phys. Rev. Lett., 83, 1999","work_id":"c4e854f5-0abb-4b7e-83bc-95b7af6391da","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"title":"Cohen.Time-frequency analysis","work_id":"8a54297e-793f-431f-9240-029afb2be1f9","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"title":"L. Cohen. Uncertainty principles of the short-time fourier transform. InAdvanced Signal Processing Algorithms, volume 2563, pages 80–90. SPIE, 1995","work_id":"e75d9ad6-f771-45b9-98c1-a278d32b8165","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"E. Cordero, M. de Gosson, and F. Nicola. On the reduction of the interferences in the Born-Jordan distribution.Appl. Comput. Harmon. Anal., 44(2), 2018","work_id":"4cae2b28-3d2d-4bc9-a3b8-1642c1c664d1","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"a1d628225e0de4183680a62a8e9844d5b16e46051ac8f6f6732da876da6df026","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"ac965098c8efd055c591752ad55feafc5bc5a72dc9b5ad2f4f8cd4100ca8452d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}