{"id":"https://openalex.org/W7224107381","doi":"https://doi.org/10.48550/arxiv.2610.35882","title":"A Counterexample to a Bourgain-Brezis Sum-Space Estimate","display_name":"A Counterexample to a Bourgain-Brezis Sum-Space Estimate","publication_year":2026,"publication_date":"2026-09-04","ids":{"openalex":"https://openalex.org/W7224107381","doi":"https://doi.org/10.48550/arxiv.2610.35882"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.35882","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35882","pdf_url":"https://arxiv.org/pdf/2610.35882","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.35882","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037905796","display_name":"Chenye Zhang","orcid":"https://orcid.org/0009-0008-4917-3726","observed_orcids":["https://orcid.org/0009-0008-4917-3726"]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhang, Chenye","raw_affiliation_strings":["Peking University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Peking University","institution_ids":["https://openalex.org/I20231570"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5037905796"],"corresponding_institution_ids":["https://openalex.org/I20231570"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11049","display_name":"Advanced Harmonic Analysis Research","score":0.9218999743461609,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11049","display_name":"Advanced Harmonic Analysis Research","score":0.9218999743461609,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.04569999873638153,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.01140000019222498,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.7120000123977661},{"id":"https://openalex.org/keywords/sobolev-spaces","display_name":"Sobolev spaces","score":0.7250000238418579},{"id":"https://openalex.org/keywords/torus","display_name":"torus","score":0.39399999380111694}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8741000294685364},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.8299999833106995},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.704200029373169},{"id":"https://openalex.org/C63356602","wikidata":"https://www.wikidata.org/wiki/Q1986013","display_name":"Lacunary function","level":2,"score":0.5322999954223633},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4984000027179718},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4936999976634979},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.448199987411499},{"id":"https://openalex.org/C118733216","wikidata":"https://www.wikidata.org/wiki/Q332504","display_name":"Absolute continuity","level":2,"score":0.4153999984264374},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4099000096321106},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.40070000290870667},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.38510000705718994},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3799999952316284},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.36899998784065247},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.3619000017642975},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.2985000014305115},{"id":"https://openalex.org/C43994743","wikidata":"https://www.wikidata.org/wiki/Q516776","display_name":"Measurable function","level":3,"score":0.2881999909877777},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2854999899864197},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.2815000116825104},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.2759999930858612},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.27320000529289246},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.2623000144958496},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26190000772476196},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.25850000977516174},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.35882","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35882","pdf_url":"https://arxiv.org/pdf/2610.35882","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.35882","doi":"https://doi.org/10.48550/arxiv.2610.35882","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.35882","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.35882","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35882","pdf_url":"https://arxiv.org/pdf/2610.35882","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224107381.pdf","grobid_xml":"https://content.openalex.org/works/W7224107381.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1571303534","https://openalex.org/W1970620614","https://openalex.org/W2011095781","https://openalex.org/W2018540675","https://openalex.org/W2044630309","https://openalex.org/W2066518895","https://openalex.org/W2080217674","https://openalex.org/W2094589439","https://openalex.org/W2155225757","https://openalex.org/W2945333717","https://openalex.org/W3105651316","https://openalex.org/W4205807684","https://openalex.org/W4246195737","https://openalex.org/W4298356938","https://openalex.org/W4388721859","https://openalex.org/W4399759583","https://openalex.org/W7156771585","https://openalex.org/W7197030808","https://openalex.org/W7202160863"],"related_works":["https://openalex.org/W4387915874","https://openalex.org/W757291033","https://openalex.org/W7100641567","https://openalex.org/W2158437858","https://openalex.org/W4298145581","https://openalex.org/W2888998067","https://openalex.org/W3085947989","https://openalex.org/W7197030808","https://openalex.org/W2292537371","https://openalex.org/W4394665369"],"abstract_inverted_index":{"A":[0,128],"question":[1],"of":[2,19,32,77,84,106],"Bourgain":[3],"and":[4,67,79],"Brezis,":[5,13],"later":[6],"restated":[7],"as":[8],"Open":[9],"Problem":[10],"7.1":[11],"by":[12,26],"asks":[14],"whether":[15],"the":[16,27,71,82,85,90,99,126],"$H^{1/2}+W^{1,1}$":[17,142],"norm":[18,31],"a":[20,55,103,107,112],"mean-zero":[21,58],"scalar":[22],"function":[23,59],"is":[24,94],"controlled":[25],"ambient":[28,91],"$H^{-1/2}+L^1$":[29],"sum":[30],"its":[33],"gradient.":[34],"We":[35],"show":[36],"that":[37,64],"this":[38],"estimate":[39],"fails":[40],"in":[41],"every":[42,49],"dimension":[43],"$N\\geq":[44],"2$.":[45],"More":[46],"precisely,":[47],"for":[48],"integer":[50],"$K\\geq":[51],"16$,":[52],"we":[53],"construct":[54],"smooth":[56,104],"real-valued":[57],"$\u03a6_K$":[60],"on":[61,111],"$\\mathbb{T}^N$":[62],"such":[63],"$|\\nabla\u03a6_K|{H^{-1/2}(\\mathbb{T}^N)+L^1(\\mathbb{T}^N)}\\leq":[65],"C$":[66],"$|\u03a6_K|{H^{1/2}(\\mathbb{T}^N)+W^{1,1}(\\mathbb{T}^N)}\\geq":[68],"cK^{1/4}-C$,":[69],"where":[70],"absolute":[72],"constants":[73],"$c,C>0$":[74],"are":[75],"independent":[76],"$N$":[78],"$K$.":[80],"Equivalently,":[81],"ratio":[83],"gradient-constrained":[86],"decomposition":[87,92],"cost":[88,93],"to":[89,102],"unbounded.":[95],"The":[96],"construction":[97],"applies":[98],"Hodge":[100],"projection":[101],"approximation":[105],"normal":[108],"vector":[109],"measure":[110],"codimension-one":[113],"sheet.":[114,127],"Its":[115],"potential":[116],"records":[117],"lacunary":[118],"partial":[119],"sums":[120],"at":[121],"geometrically":[122],"separated":[123],"distances":[124],"from":[125],"scale-path":[129],"lower":[130],"bound,":[131],"combined":[132],"with":[133],"an":[134],"odd":[135],"critical":[136],"Hardy":[137],"inequality,":[138],"excludes":[139],"uniformly":[140],"bounded":[141],"decompositions.":[143]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-29T00:00:00"}
