{"id":"https://openalex.org/W7204970926","doi":"https://doi.org/10.48550/arxiv.2608.29218","title":"A Littlewood-Type Theorem for Harmonic Functions in Euclidean Half-Spaces","display_name":"A Littlewood-Type Theorem for Harmonic Functions in Euclidean Half-Spaces","publication_year":2026,"publication_date":"2026-08-29","ids":{"openalex":"https://openalex.org/W7204970926","doi":"https://doi.org/10.48550/arxiv.2608.29218"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.29218","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29218","pdf_url":"https://arxiv.org/pdf/2608.29218","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/2608.29218","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5066713005","display_name":"F. Di Biase","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Di Biase, F.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5148843285","display_name":"H. Gratien","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210087849","display_name":"University of Rwanda","ror":"https://ror.org/00286hs46","country_code":"RW","type":"education","lineage":["https://openalex.org/I4210087849"]}],"countries":["RW"],"is_corresponding":false,"raw_author_name":"Gratien, H.","raw_affiliation_strings":["Department of Mathematics University of Rwanda KN 67 Nyarugenge , Kigali , 3900 Kigali , Rwanda"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics University of Rwanda KN 67 Nyarugenge , Kigali , 3900 Kigali , Rwanda","institution_ids":["https://openalex.org/I4210087849"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5148893310","display_name":"O. Svensson","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I102134673","display_name":"Link\u00f6ping University","ror":"https://ror.org/05ynxx418","country_code":"SE","type":"education","lineage":["https://openalex.org/I102134673"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"Svensson, O.","raw_affiliation_strings":["Department of Science and Technology Link\u00f6ping University SE-60174 Norrk\u00f6ping , Sweden"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Science and Technology Link\u00f6ping University SE-60174 Norrk\u00f6ping , Sweden","institution_ids":["https://openalex.org/I102134673"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"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":0.986299991607666,"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":0.986299991607666,"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/T10884","display_name":"Holomorphic and Operator Theory","score":0.003100000089034438,"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/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.0020000000949949026,"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/bounded-harmonic-functions","display_name":"bounded harmonic functions","score":0.9580000042915344}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7789999842643738},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7343000173568726},{"id":"https://openalex.org/C627467","wikidata":"https://www.wikidata.org/wiki/Q599027","display_name":"Harmonic function","level":2,"score":0.7005000114440918},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5626999735832214},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5091999769210815},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5023000240325928},{"id":"https://openalex.org/C53688716","wikidata":"https://www.wikidata.org/wiki/Q5659270","display_name":"Harmonic measure","level":3,"score":0.46389999985694885},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.46369999647140503},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4578000009059906},{"id":"https://openalex.org/C27142425","wikidata":"https://www.wikidata.org/wiki/Q1139334","display_name":"Almost everywhere","level":2,"score":0.4415000081062317},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.39649999141693115},{"id":"https://openalex.org/C92928283","wikidata":"https://www.wikidata.org/wiki/Q4949981","display_name":"Bounded deformation","level":4,"score":0.39469999074935913},{"id":"https://openalex.org/C98343798","wikidata":"https://www.wikidata.org/wiki/Q1790208","display_name":"Curvilinear coordinates","level":2,"score":0.3758000135421753},{"id":"https://openalex.org/C191948623","wikidata":"https://www.wikidata.org/wiki/Q1307800","display_name":"Unit sphere","level":2,"score":0.33820000290870667},{"id":"https://openalex.org/C29373527","wikidata":"https://www.wikidata.org/wiki/Q753658","display_name":"Subharmonic function","level":2,"score":0.33640000224113464},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.33570000529289246},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.31779998540878296},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.30059999227523804},{"id":"https://openalex.org/C205979905","wikidata":"https://www.wikidata.org/wiki/Q215084","display_name":"Analytic function","level":2,"score":0.2930000126361847},{"id":"https://openalex.org/C43994743","wikidata":"https://www.wikidata.org/wiki/Q516776","display_name":"Measurable function","level":3,"score":0.29100000858306885},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.27790001034736633},{"id":"https://openalex.org/C47177299","wikidata":"https://www.wikidata.org/wiki/Q2390323","display_name":"Unit disk","level":2,"score":0.2770000100135803},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C131770355","wikidata":"https://www.wikidata.org/wiki/Q876215","display_name":"Harmonic analysis","level":2,"score":0.26080000400543213},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.258899986743927},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.2563000023365021}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.29218","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29218","pdf_url":"https://arxiv.org/pdf/2608.29218","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.2608.29218","doi":"https://doi.org/10.48550/arxiv.2608.29218","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.29218","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:2608.29218","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29218","pdf_url":"https://arxiv.org/pdf/2608.29218","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":[{"id":"https://openalex.org/F4320307875","display_name":"Styrelsen f\u00f6r Internationellt Utvecklingssamarbete","ror":"https://ror.org/01fn7me06"},{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204970926.pdf","grobid_xml":"https://content.openalex.org/works/W7204970926.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4416378462"],"related_works":["https://openalex.org/W4416378462","https://openalex.org/W1984650967","https://openalex.org/W799496016","https://openalex.org/W2035333765","https://openalex.org/W2002664580","https://openalex.org/W7099361014","https://openalex.org/W2211917569","https://openalex.org/W2005815764","https://openalex.org/W1500031248","https://openalex.org/W4396799369"],"abstract_inverted_index":{"In":[0,147],"1927":[1],"J.E.":[2],"Littlewood":[3,132,202],"proved":[4,81],"that,":[5],"for":[6,33,155,163],"bounded":[7,101,156],"harmonic":[8,102,157],"functions":[9,158],"on":[10,17],"the":[11,14,18,42,53,82,125,171,176,192,198,204,224],"unit":[12,193],"disc,":[13],"Fatou-type":[15,94],"result,":[16],"existence":[19,83],"of":[20,36,49,72,84,88,127,165,179,187,201],"almost":[21],"everywhere":[22],"boundary":[23,55,109],"values":[24],"through":[25],"approach":[26,38,142,167,206],"regions":[27,39,143,168,207],"that":[28,40,135,222,226],"are":[29,145],"nontangential,":[30],"will":[31],"fail":[32],"any":[34],"system":[35],"tangential":[37,89,141,166,205],"has":[41,123],"following":[43],"two":[44],"additional":[45],"properties:":[46],"It":[47],"consists":[48],"curves":[50],"ending":[51],"at":[52,214],"various":[54],"points,":[56],"and":[57,67,75,78,129],"it":[58],"is":[59,175,191],"rotationally":[60,85],"invariant.":[61],"However,":[62],"in":[63,159,196],"1984":[64],"A.":[65],"Nagel":[66],"E.M.":[68],"Stein,":[69],"elaborating":[70],"results":[71,200],"Rudin":[73,77],"(1979)":[74],"Nagel,":[76],"Shapiro":[79],"(1982),":[80],"invariant":[86],"systems":[87,164],"sequences":[90],"along":[91,98],"which":[92,99,144,169],"a":[93,131,152,184,218],"result":[95,115,122,178,186],"holds,":[96],"i.e.,":[97],"every":[100],"function":[103],"converges":[104],"a.e.":[105],"to":[106,116,140,210,216],"its":[107],"nontangential":[108],"values.":[110],"Moreover,":[111],"they":[112,227],"extended":[113],"their":[114],"higher-dimensional":[117,160],"Euclidean":[118,161],"half-spaces.":[119],"The":[120],"Nagel-Stein":[121],"prompted":[124],"question":[126],"formulating":[128],"proving":[130],"type":[133],"theorem":[134,154],"can":[136],"also":[137],"be":[138,211,229],"applied":[139],"sequential.":[146,230],"this":[148,180],"paper":[149],"we":[150],"prove":[151],"Littlewood-type":[153],"half-spaces,":[162],"includes":[170],"sequential":[172],"ones.":[173],"This":[174],"first":[177],"kind,":[181],"apart":[182],"from":[183],"recent":[185],"ours":[188],"whose":[189],"setting":[190],"disc.":[194],"Indeed,":[195],"all":[197],"other":[199],"type,":[203],"were":[208],"required":[209],"curvilinear":[212],"or":[213],"least":[215],"possess":[217],"certain":[219],"topological":[220],"property":[221],"excluded":[223],"possibility":[225],"could":[228]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-02T00:00:00"}