{"id":"https://openalex.org/W7214601900","doi":"https://doi.org/10.48550/arxiv.2609.30949","title":"A unified approach to nonlinear Stein theorems","display_name":"A unified approach to nonlinear Stein theorems","publication_year":2026,"publication_date":"2026-09-25","ids":{"openalex":"https://openalex.org/W7214601900","doi":"https://doi.org/10.48550/arxiv.2609.30949"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.30949","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30949","pdf_url":"https://arxiv.org/pdf/2609.30949","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.30949","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014115108","display_name":"Arka Mallick","orcid":"https://orcid.org/0000-0001-9870-2112","observed_orcids":["https://orcid.org/0000-0001-9870-2112"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mallick, Arka","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5059842904","display_name":"Swarnendu Sil","orcid":"https://orcid.org/0000-0002-9417-3345","observed_orcids":["https://orcid.org/0000-0002-9417-3345"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sil, Swarnendu","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":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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9657999873161316,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9657999873161316,"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":0.016699999570846558,"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":0.004100000020116568,"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/variable-exponent","display_name":"variable exponent","score":0.925000011920929},{"id":"https://openalex.org/keywords/heisenberg-group","display_name":"Heisenberg group","score":0.949999988079071},{"id":"https://openalex.org/keywords/sub-laplacian","display_name":"sub-Laplacian","score":0.8949999809265137},{"id":"https://openalex.org/keywords/first-derivative","display_name":"first derivative","score":0.5759999752044678},{"id":"https://openalex.org/keywords/mean-oscillation","display_name":"mean oscillation","score":0.6930000185966492},{"id":"https://openalex.org/keywords/homogeneous-dimension","display_name":"homogeneous dimension","score":0.6460000276565552},{"id":"https://openalex.org/keywords/weak-solutions","display_name":"weak solutions","score":0.6439999938011169},{"id":"https://openalex.org/keywords/subelliptic-equations","display_name":"subelliptic equations","score":0.6539999842643738},{"id":"https://openalex.org/keywords/elliptic-equations","display_name":"elliptic equations","score":0.5630000233650208}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8410000205039978},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.758899986743927},{"id":"https://openalex.org/C127519595","wikidata":"https://www.wikidata.org/wiki/Q1601337","display_name":"Heisenberg group","level":2,"score":0.683899998664856},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.644599974155426},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5986999869346619},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5260000228881836},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.5212000012397766},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4846999943256378},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.4724999964237213},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4569999873638153},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.39640000462532043},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.38760000467300415},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3725000023841858},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3587000072002411},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.3531999886035919},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.3384000062942505},{"id":"https://openalex.org/C164154869","wikidata":"https://www.wikidata.org/wiki/Q901927","display_name":"Critical exponent","level":3,"score":0.32120001316070557},{"id":"https://openalex.org/C94906876","wikidata":"https://www.wikidata.org/wiki/Q1132952","display_name":"Homogeneous function","level":3,"score":0.32109999656677246},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.31630000472068787},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3107999861240387},{"id":"https://openalex.org/C10784197","wikidata":"https://www.wikidata.org/wiki/Q233858","display_name":"Non-Euclidean geometry","level":3,"score":0.27219998836517334},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.257099986076355}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.30949","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30949","pdf_url":"https://arxiv.org/pdf/2609.30949","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.2609.30949","doi":"https://doi.org/10.48550/arxiv.2609.30949","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.30949","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.30949","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30949","pdf_url":"https://arxiv.org/pdf/2609.30949","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":[{"id":"https://openalex.org/G1488853880","display_name":null,"funder_award_id":"TPN\u2010700661","funder_id":"https://openalex.org/F4320320719","funder_display_name":"Department of Science and Technology, Ministry of Science and Technology, India"},{"id":"https://openalex.org/G3192681547","display_name":null,"funder_award_id":"MTR/2023/000885","funder_id":"https://openalex.org/F4320334771","funder_display_name":"Science and Engineering Research Board"}],"funders":[{"id":"https://openalex.org/F4320306227","display_name":"Arthritis National Research Foundation","ror":"https://ror.org/03abbex33"},{"id":"https://openalex.org/F4320310071","display_name":"Indian Institute of Science","ror":"https://ror.org/04dese585"},{"id":"https://openalex.org/F4320320719","display_name":"Department of Science and Technology, Ministry of Science and Technology, India","ror":"https://ror.org/0101xrq71"},{"id":"https://openalex.org/F4320334771","display_name":"Science and Engineering Research Board","ror":"https://ror.org/03ffdsr55"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214601900.pdf","grobid_xml":"https://content.openalex.org/works/W7214601900.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4403821392","https://openalex.org/W2312053986","https://openalex.org/W2058357345","https://openalex.org/W7096580149","https://openalex.org/W4388888577","https://openalex.org/W4308438503","https://openalex.org/W2029744607","https://openalex.org/W2795506988","https://openalex.org/W7214635990","https://openalex.org/W2964226705"],"abstract_inverted_index":{"We":[0,19,141],"study":[1],"regularity":[2],"for":[3,24,37,132],"the":[4,25,77,109,118,123,133,143,168,174,179],"variable":[5],"exponent":[6,78],"quasilinear":[7],"$\\mathfrak{p}\\left(x\\right)$-Laplace":[8],"type":[9],"equation":[10],"on":[11,158],"domains":[12],"in":[13,61,92,148,173],"Heisenberg":[14,119],"groups":[15],"and":[16,76,87,102,127,137,145,155,160,164],"Euclidean":[17,124,153,175],"spaces.":[18],"establish":[20],"borderline":[21],"continuity":[22],"estimates":[23],"appropriate":[26],"first":[27],"order":[28],"derivatives":[29],"of":[30,45,111],"solutions.":[31],"More":[32],"precisely,":[33],"we":[34],"show":[35],"that":[36],"any":[38],"weak":[39],"solution":[40],"$u":[41],"\\in":[42,98],"\\mathbb{E}W^{1,":[43],"\\mathfrak{p}(\\cdot)}\\left(\u03a9\\right)$":[44],"\\begin{align*}":[46],"\\operatorname{div}_{\\mathbb{E}}":[47],"\\left(":[48],"\\mathfrak{a}(x)":[49],"\\lvert":[50],"\\nabla_{\\mathbb{E}}":[51,54],"u":[52,55],"\\rvert^{\\mathfrak{p}\\left(x\\right)-2}":[53],"\\right)":[56],"=":[57],"f":[58],"\\qquad":[59],"\\text{":[60],"}":[62],"\u03a9,":[63],"\\end{align*}":[64],"where":[65],"$\u03a9\\subset":[66],"\\mathbb{E}^{n}$,":[67],"$\\mathfrak{a}$":[68,159],"is":[69,81,90,116],"a":[70,149],"uniformly":[71,82],"positive":[72],"bounded":[73,83],"scalar":[74],"function":[75,79],"$\\mathfrak{p}$":[80,161],"away":[84],"from":[85],"$1$":[86],"$\\infty,$":[88],"$\\nabla_{\\mathbb{E}}u$":[89],"continuous":[91],"$\u03a9$":[93],"as":[94,96],"soon":[95],"$f":[97],"L^{\\left(":[99],"Q_{\\mathbb{E}},":[100],"1\\right)}\\left(\u03a9\\right)$":[101],"$\\mathfrak{a},":[103],"\\mathfrak{p}$":[104],"satisfies":[105],"some":[106],"conditions":[107,157,171,182],"regarding":[108],"summability":[110],"their":[112],"mean-oscillations.":[113],"Here":[114],"$\\mathbb{E}^{n}$":[115],"either":[117],"group":[120],"$\\mathbb{H}_{n}$":[121],"or":[122],"space":[125],"$\\mathbb{R}^{n}$":[126],"$\\operatorname{div}_{\\mathbb{E}}$,":[128],"$\\nabla_{\\mathbb{E}}$,":[129],"$Q_{\\mathbb{E}}$":[130],"stands":[131],"corresponding":[134],"divergence,":[135],"gradient":[136],"homogeneous":[138],"dimension,":[139],"respectively.":[140],"treat":[142],"elliptic":[144],"subelliptic":[146],"cases":[147],"unified":[150],"manner":[151],"using":[152],"techniques":[154],"our":[156,184],"are":[162],"new":[163],"weaker":[165],"than":[166],"all":[167,178],"known":[169,180],"sufficient":[170,181],"even":[172],"case.":[176],"However,":[177],"imply":[183],"conditions,":[185],"achieving":[186],"yet":[187],"another":[188],"unification.":[189]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-29T00:00:00"}
