{"id":"https://openalex.org/W7204253155","doi":"https://doi.org/10.48550/arxiv.2608.24106","title":"Weak-type characterizations of Sobolev and bounded variation spaces on metric measure spaces","display_name":"Weak-type characterizations of Sobolev and bounded variation spaces on metric measure spaces","publication_year":2026,"publication_date":"2026-08-25","ids":{"openalex":"https://openalex.org/W7204253155","doi":"https://doi.org/10.48550/arxiv.2608.24106"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.24106","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24106","pdf_url":"https://arxiv.org/pdf/2608.24106","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2608.24106","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5134202256","display_name":"Tuomas P. Hyt\u00f6nen","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hyt\u00f6nen, Tuomas P.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5125665134","display_name":"Dachun Yang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yang, Dachun","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5121895677","display_name":"Wen Yuan","orcid":"https://orcid.org/0000-0002-4072-245X","observed_orcids":["https://orcid.org/0000-0002-4072-245X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yuan, Wen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5093976384","display_name":"Yirui Zhao","orcid":"https://orcid.org/0009-0001-7875-2574","observed_orcids":["https://orcid.org/0009-0001-7875-2574"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhao, Yirui","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":5,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.7705000042915344,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.7705000042915344,"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/T11049","display_name":"Advanced Harmonic Analysis Research","score":0.20260000228881836,"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":0.011099999770522118,"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/sobolev-spaces","display_name":"Sobolev spaces","score":0.9380000233650208},{"id":"https://openalex.org/keywords/functions-of-bounded-variation","display_name":"functions of bounded variation","score":0.8529999852180481},{"id":"https://openalex.org/keywords/metric-measure-spaces","display_name":"metric measure spaces","score":0.9269999861717224},{"id":"https://openalex.org/keywords/poincare-inequality","display_name":"Poincar\u00e9 inequality","score":0.8930000066757202},{"id":"https://openalex.org/keywords/homogeneous-sobolev-spaces","display_name":"homogeneous Sobolev spaces","score":0.8450000286102295},{"id":"https://openalex.org/keywords/mean-oscillation","display_name":"mean oscillation","score":0.8579999804496765},{"id":"https://openalex.org/keywords/norm-equivalence","display_name":"norm equivalence","score":0.3930000066757202}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8824999928474426},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.7318999767303467},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6413999795913696},{"id":"https://openalex.org/C27851653","wikidata":"https://www.wikidata.org/wiki/Q829653","display_name":"Bounded variation","level":2,"score":0.5453000068664551},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.5085999965667725},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5080000162124634},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.4896000027656555},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.489300012588501},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.426800012588501},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39399999380111694},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.3628999888896942},{"id":"https://openalex.org/C6183050","wikidata":"https://www.wikidata.org/wiki/Q3058180","display_name":"Interpolation space","level":4,"score":0.3601999878883362},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.36000001430511475},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3499000072479248},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.3287000060081482},{"id":"https://openalex.org/C27142425","wikidata":"https://www.wikidata.org/wiki/Q1139334","display_name":"Almost everywhere","level":2,"score":0.32350000739097595},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.3163999915122986},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30809998512268066},{"id":"https://openalex.org/C129301229","wikidata":"https://www.wikidata.org/wiki/Q741865","display_name":"Uniform continuity","level":3,"score":0.30390000343322754},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.3034000098705292},{"id":"https://openalex.org/C185644265","wikidata":"https://www.wikidata.org/wiki/Q3711848","display_name":"Sobolev inequality","level":3,"score":0.3010999858379364},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.30079999566078186},{"id":"https://openalex.org/C2776551381","wikidata":"https://www.wikidata.org/wiki/Q6680542","display_name":"Lorentz space","level":3,"score":0.29670000076293945},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.2816999852657318},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.2736000120639801},{"id":"https://openalex.org/C43994743","wikidata":"https://www.wikidata.org/wiki/Q516776","display_name":"Measurable function","level":3,"score":0.2685999870300293},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.2644999921321869},{"id":"https://openalex.org/C2777044963","wikidata":"https://www.wikidata.org/wiki/Q5384707","display_name":"Equivalence class (music)","level":2,"score":0.26089999079704285}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.24106","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24106","pdf_url":"https://arxiv.org/pdf/2608.24106","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.24106","doi":"https://doi.org/10.48550/arxiv.2608.24106","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.24106","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.24106","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24106","pdf_url":"https://arxiv.org/pdf/2608.24106","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G5710692465","display_name":null,"funder_award_id":"2253200028","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G5884157002","display_name":null,"funder_award_id":"12431006","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6737089312","display_name":null,"funder_award_id":"#12371093","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321106","display_name":"Ministry of Education of the People's Republic of China","ror":"https://ror.org/01mv9t934"},{"id":"https://openalex.org/F4320321394","display_name":"Aalto-Yliopisto","ror":"https://ror.org/020hwjq30"},{"id":"https://openalex.org/F4320321432","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204253155.pdf","grobid_xml":"https://content.openalex.org/works/W7204253155.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4415066461","https://openalex.org/W4415194420"],"related_works":["https://openalex.org/W2790913538","https://openalex.org/W4284890478","https://openalex.org/W2528055655","https://openalex.org/W2372062312","https://openalex.org/W1573529790","https://openalex.org/W1575018994","https://openalex.org/W4394948205","https://openalex.org/W2963776936","https://openalex.org/W7096451024","https://openalex.org/W2061732609"],"abstract_inverted_index":{"Given":[0],"a":[1,9,30,58,75,118,132,211],"complete":[2],"doubling":[3],"metric":[4],"measure":[5],"space":[6,19,23,166],"$(X,\u03c1,\u03bc)$":[7],"supporting":[8],"Poincar\u00e9":[10,35],"inequality,":[11],"we":[12,140,181],"prove":[13,141],"weak-type":[14,59],"characterizations":[15],"of":[16,24,26,41,57,69,81,98,194,237],"the":[17,22,38,55,79,84,95,111,117,163,171,191,197,202,217,220,234],"Sobolev":[18,86,165],"$\\dot{W}^{1,p}(\u03bc)$":[20,167],"and":[21,145,184,188,215],"functions":[25],"bounded":[27,212],"variation,":[28],"achieving":[29],"full":[31],"analogy":[32],"in":[33,83,128],"general":[34],"spaces":[36],"with":[37],"Euclidean":[39],"results":[40],"Brezis":[42],"et":[43,101],"al.":[44,102],"[Anal.":[45],"PDE":[46],"17":[47],"(2024),":[48],"943-979].":[49],"The":[50],"main":[51],"novelty":[52],"is":[53,131,168,190,233],"that":[54],"finiteness":[56],"norm,":[60],"which":[61],"only":[62],"refers":[63],"to":[64,219],"differences":[65],"or":[66,87,226],"mean":[67,192],"oscillations":[68],"$f$":[70,82,195],"without":[71],"assuming":[72],"any":[73,150,178],"smoothness":[74],"priori,":[76],"already":[77],"guarantees":[78],"membership":[80],"relevant":[85],"BV":[88],"space.":[89],"This":[90],"distinguishes":[91],"our":[92,129],"contribution":[93],"from":[94],"recent":[96],"work":[97],"F.":[99],"Dai":[100],"[Adv.":[103],"Math.":[104],"502":[105],"(2026),":[106],"Paper":[107],"No.":[108],"111153],":[109],"where":[110,162,231],"related":[112],"norm-equivalence":[113],"was":[114],"obtained":[115],"under":[116],"priori":[119],"Lipschitz":[120],"assumption":[121],"on":[122,196],"$f$.":[123],"A":[124],"key":[125],"intermediate":[126],"step":[127],"approach":[130],"new":[133],"localized":[134],"Bourgain-Brezis-Mironescu":[135],"type":[136],"characterization.":[137],"More":[138],"precisely,":[139],"that,":[142],"if":[143],"$p\\in(1,\\infty)$":[144],"$\u03b3\\in\\mathbb":[146],"R\\setminus\\{0\\}$,":[147],"then,":[148],"for":[149,177,224,229],"$f\\in":[151],"L^1_{\\mathrm{loc}}(\u03bc)$,":[152],"\\begin{equation*}\\tag{$*$}":[153],"\\|f\\|_{\\dot":[154],"W^{1,p}(\u03bc)}":[155],"\\sim":[156],"\\|\u03c1^{-1}\u03d5^{-\u03b3}F\\|_{L^{p,\\infty}(\u03d5^{\u03b3p}V^{-1})},":[157],"\\qquad":[158],"F\\in\\{\u0394f,m_f\\},\\quad":[159],"\u03d5\\in\\{\u03c1,V\\},":[160],"\\end{equation*}":[161],"homogeneous":[164],"defined":[169],"by":[170,210],"minimal":[172],"$p$-weak":[173],"upper":[174],"gradient":[175],"and,":[176],"$x,y\\in":[179],"X$,":[180],"denote":[182],"$V(x,y):=\u03bc(B(x,\u03c1(x,y)))$":[183],"$\u0394f(x,y):=|f(x)":[185],"-":[186],"f(y)|$,":[187],"$m_f(x,y)$":[189],"oscillation":[193],"ball":[198],"$B(x,\u03c1(x,y))$.":[199],"For":[200],"$p=1$,":[201],"equivalence":[203],"$(*)$":[204],"holds":[205],"after":[206],"replacing":[207],"$\\|f\\|_{\\dot":[208],"W^{1,1}(\u03bc)}$":[209],"variation":[213],"norm":[214],"restricting":[216],"parameters":[218],"optimal":[221],"ranges":[222],"$\u03b3\\in(-\\infty,-1)\\cup(0,\\infty)$":[223],"$\u03d5=\u03c1$":[225],"$\u03b3\\in":[227],"(-\\infty,-\\frac1d)\\cup(0,\\infty)$":[228],"$\u03d5=V$,":[230],"$d\\in(0,\\infty)$":[232],"lower":[235],"dimension":[236],"$X$.":[238]},"counts_by_year":[{"year":2026,"cited_by_count":5}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-27T00:00:00"}
