{"id":"https://openalex.org/W7212185047","doi":"https://doi.org/10.48550/arxiv.2609.10208","title":"The measure contraction property on Grushin spaces","display_name":"The measure contraction property on Grushin spaces","publication_year":2026,"publication_date":"2026-09-09","ids":{"openalex":"https://openalex.org/W7212185047","doi":"https://doi.org/10.48550/arxiv.2609.10208"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.10208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10208","pdf_url":"https://arxiv.org/pdf/2609.10208","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/2609.10208","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039211301","display_name":"Michael Albert","orcid":"https://orcid.org/0000-0002-8478-3043","observed_orcids":["https://orcid.org/0000-0002-8478-3043"]},"institutions":[{"id":"https://openalex.org/I142637625","display_name":"Okinawa Institute of Science and Technology Graduate University","ror":"https://ror.org/02qg15b79","country_code":"JP","type":"education","lineage":["https://openalex.org/I142637625"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Albert, Michael","raw_affiliation_strings":["Analysis and PDE Unit , Okinawa Institute of Science and Technology , 1919-1 Tancha , Onna-son , Kunigami-gun , Okinawa 904-0495 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Analysis and PDE Unit , Okinawa Institute of Science and Technology , 1919-1 Tancha , Onna-son , Kunigami-gun , Okinawa 904-0495 , Japan","institution_ids":["https://openalex.org/I142637625"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5022914554","display_name":"Samu\u00ebl Borza","orcid":"https://orcid.org/0000-0001-7635-1294","observed_orcids":["https://orcid.org/0000-0001-7635-1294"]},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Borza, Samu\u00ebl","raw_affiliation_strings":["Faculty of Mathematics , University of Vienna , Oskar-Morgenstern-Platz 1 , 1090 Vienna , Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics , University of Vienna , Oskar-Morgenstern-Platz 1 , 1090 Vienna , Austria","institution_ids":["https://openalex.org/I129774422"]}]}],"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9990000128746033,"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":0.9990000128746033,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.000699999975040555,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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":0.0,"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/metric-measure-spaces","display_name":"metric measure spaces","score":0.7979999780654907},{"id":"https://openalex.org/keywords/sub-riemannian-geometry","display_name":"sub-Riemannian geometry","score":0.8379999995231628},{"id":"https://openalex.org/keywords/sharp-constants","display_name":"sharp constants","score":0.671999990940094},{"id":"https://openalex.org/keywords/curvature-preserving-deformations","display_name":"curvature-preserving deformations","score":0.4440000057220459}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8216999769210815},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.7527999877929688},{"id":"https://openalex.org/C163415756","wikidata":"https://www.wikidata.org/wiki/Q126473","display_name":"Contraction (grammar)","level":2,"score":0.6543999910354614},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.6353999972343445},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5662999749183655},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5546000003814697},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5295000076293945},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.5055999755859375},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.46619999408721924},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.4011000096797943},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.38179999589920044},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.3736000061035156},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3578000068664551},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33709999918937683},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.313400000333786},{"id":"https://openalex.org/C12871628","wikidata":"https://www.wikidata.org/wiki/Q7598372","display_name":"Standard probability space","level":2,"score":0.305400013923645},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.28049999475479126},{"id":"https://openalex.org/C172294467","wikidata":"https://www.wikidata.org/wiki/Q305936","display_name":"Lp space","level":3,"score":0.2612000107765198}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.10208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10208","pdf_url":"https://arxiv.org/pdf/2609.10208","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.2609.10208","doi":"https://doi.org/10.48550/arxiv.2609.10208","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.10208","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:2609.10208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10208","pdf_url":"https://arxiv.org/pdf/2609.10208","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212185047.pdf","grobid_xml":"https://content.openalex.org/works/W7212185047.grobid-xml"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W1817625155","https://openalex.org/W1883771592","https://openalex.org/W1902153087","https://openalex.org/W1941013316","https://openalex.org/W1968781546","https://openalex.org/W1972088038","https://openalex.org/W1978419408","https://openalex.org/W1978889356","https://openalex.org/W1979858013","https://openalex.org/W2007338883","https://openalex.org/W2024150398","https://openalex.org/W2026405417","https://openalex.org/W2026841652","https://openalex.org/W2045049746","https://openalex.org/W2055531847","https://openalex.org/W2078255209","https://openalex.org/W2097011226","https://openalex.org/W2108819385","https://openalex.org/W2145507879","https://openalex.org/W2146640514","https://openalex.org/W2157490453","https://openalex.org/W2270433959","https://openalex.org/W2287751836","https://openalex.org/W2320112368","https://openalex.org/W2331387868","https://openalex.org/W2336023452","https://openalex.org/W2518161628","https://openalex.org/W2585847086","https://openalex.org/W2936930064","https://openalex.org/W3045786417","https://openalex.org/W3094993144","https://openalex.org/W3153684863","https://openalex.org/W4210956211","https://openalex.org/W4229706427","https://openalex.org/W4248500611","https://openalex.org/W4327981257","https://openalex.org/W4364372472","https://openalex.org/W4413076727","https://openalex.org/W4415198139","https://openalex.org/W4416878741","https://openalex.org/W4417448186","https://openalex.org/W7151233470","https://openalex.org/W7204870080"],"related_works":["https://openalex.org/W2076825508","https://openalex.org/W2800761877","https://openalex.org/W3094993144","https://openalex.org/W7098780472","https://openalex.org/W2045049746","https://openalex.org/W3128873093","https://openalex.org/W2898401123","https://openalex.org/W4403706528","https://openalex.org/W4206283237","https://openalex.org/W2768872545"],"abstract_inverted_index":{"We":[0,40,55],"determine":[1],"the":[2,61,87,97,111,117],"sharp":[3],"measure":[4,13,25],"contraction":[5],"exponents":[6],"of":[7,10,100,116],"two":[8],"families":[9],"Grushin-type":[11],"metric":[12],"spaces.":[14],"The":[15],"radial":[16],"Grushin":[17,119],"space":[18],"$\\mathbb{G}^{n+m}$":[19,43],"is":[20],"$\\mathbb{R}^{n}\\times\\mathbb{R}^{m}$,":[21],"equipped":[22],"with":[23,104],"Lebesgue":[24],"and":[26,30,36,47,52,67,72,76,125],"generated":[27,64],"by":[28,65],"$X_i=\\partial_{x_i}$":[29],"$Y_j=|x|\\partial_{y_j}$,":[31],"for":[32,59,93],"$1\\leq":[33,37],"i\\leq":[34],"n$":[35],"j\\leq":[38],"m$.":[39],"prove":[41],"that":[42],"satisfies":[44,69],"$\\operatorname{MCP}(K,N)$":[45,70],"if":[46,49,71,74],"only":[48,73],"$N\\geq":[50,77],"n+4m$":[51],"$K\\leq":[53],"0$.":[54],"also":[56],"show":[57],"that,":[58],"$\u03b1\\geq1$,":[60],"$\u03b1$-Grushin":[62],"plane":[63],"$X=\\partial_x$":[66],"$Y_\u03b1=|x|^\u03b1\\partial_y$":[68],"$K\\leq0$":[75],"N_\u03b1$,":[78],"where":[79],"\\[":[80],"N_\u03b1:=":[81],"1+\\max_{L>1}":[82],"\\frac{(2\u03b1+1)L}{(L-1)^{2\u03b1+1}+1}.":[83],"\\]":[84],"This":[85],"resolves":[86],"conjecture":[88],"posed":[89],"in":[90],"arXiv:2010.16350":[91],"and,":[92],"integer":[94],"$\u03b1\\geq2$,":[95],"provides":[96],"first":[98],"examples":[99],"real-analytic":[101],"sub-Riemannian":[102],"structures":[103],"noninteger":[105],"curvature":[106,113],"exponent.":[107],"Both":[108],"results":[109],"recover":[110],"known":[112],"exponent":[114],"$5$":[115],"classical":[118],"plane,":[120],"corresponding":[121],"respectively":[122],"to":[123],"$n=m=1$":[124],"$\u03b1=1$.":[126]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-11T00:00:00"}
