{"id":"https://openalex.org/W3120569918","doi":"https://doi.org/10.48550/arxiv.2101.02495","title":"Convergence over fractals for the Schr\u00f6dinger equation","display_name":"Convergence over fractals for the Schr\u00f6dinger equation","publication_year":2021,"publication_date":"2021-01-07","ids":{"openalex":"https://openalex.org/W3120569918","doi":"https://doi.org/10.48550/arxiv.2101.02495","mag":"3120569918"},"language":"en","primary_location":{"id":"mag:3120569918","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2101.02495.pdf","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2101.02495.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008270502","display_name":"Renato Luc\u00e0","orcid":"https://orcid.org/0000-0002-7983-7967","observed_orcids":["https://orcid.org/0000-0002-7983-7967"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Luc\u00e0, Renato","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5047467666","display_name":"Felipe Ponce-Vanegas","orcid":"https://orcid.org/0000-0002-1049-9752","observed_orcids":["https://orcid.org/0000-0002-1049-9752"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ponce-Vanegas, Felipe","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":2,"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":0.5396000146865845,"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":0.5396000146865845,"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/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.43309998512268066,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.009999999776482582,"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/schrodinger-equation","display_name":"Schr\u00f6dinger equation","score":0.9769999980926514},{"id":"https://openalex.org/keywords/pointwise-convergence","display_name":"pointwise convergence","score":0.878000020980835},{"id":"https://openalex.org/keywords/minimal-regularity","display_name":"minimal regularity","score":0.6520000100135803},{"id":"https://openalex.org/keywords/almost-everywhere-convergence","display_name":"almost everywhere convergence","score":0.671999990940094}],"concepts":[{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.8623834848403931},{"id":"https://openalex.org/C27156116","wikidata":"https://www.wikidata.org/wiki/Q1778098","display_name":"Pointwise convergence","level":3,"score":0.8298366069793701},{"id":"https://openalex.org/C27142425","wikidata":"https://www.wikidata.org/wiki/Q1139334","display_name":"Almost everywhere","level":2,"score":0.8238174319267273},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.7779545783996582},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.7374077439308167},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7075947523117065},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.6644458770751953},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.6403453350067139},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6198994517326355},{"id":"https://openalex.org/C64943373","wikidata":"https://www.wikidata.org/wiki/Q2651003","display_name":"Alpha (finance)","level":4,"score":0.578156054019928},{"id":"https://openalex.org/C73225184","wikidata":"https://www.wikidata.org/wiki/Q1591095","display_name":"Hausdorff measure","level":3,"score":0.5302391052246094},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.4929497241973877},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.395651251077652},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3818577826023102},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32260996103286743},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.13832280039787292},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08213087916374207},{"id":"https://openalex.org/C2777894999","wikidata":"https://www.wikidata.org/wiki/Q4781758","display_name":"Approx","level":2,"score":0.04336518049240112},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C49453240","wikidata":"https://www.wikidata.org/wiki/Q1592163","display_name":"Construct validity","level":3,"score":0.0},{"id":"https://openalex.org/C171606756","wikidata":"https://www.wikidata.org/wiki/Q506132","display_name":"Psychometrics","level":2,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"mag:3120569918","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2101.02495.pdf","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:bird.bcamath.org:20.500.11824/1240","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11824/1240","pdf_url":null,"source":{"id":"https://openalex.org/S4306401608","display_name":"BIRD (Basque Center for Applied Mathematics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I2802176441","host_organization_name":"Basque Center for Applied Mathematics","host_organization_lineage":["https://openalex.org/I2802176441"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:dnet:bird________::70f1076f9aa122fae62b193090a36100","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11824/1240","pdf_url":null,"source":{"id":"https://openalex.org/S4306402641","display_name":"LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Cient\u00edficas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4383465926","host_organization_name":"LA Referencia","host_organization_lineage":["https://openalex.org/I4383465926"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.2101.02495","doi":"https://doi.org/10.48550/arxiv.2101.02495","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2101.02495","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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"mag:3120569918","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2101.02495.pdf","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321705","display_name":"Eusko Jaurlaritza","ror":"https://ror.org/00pz2fp31"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W135001645","https://openalex.org/W1198888588","https://openalex.org/W1497523820","https://openalex.org/W1580562883","https://openalex.org/W1584239381","https://openalex.org/W1968830247","https://openalex.org/W1978662909","https://openalex.org/W2042538227","https://openalex.org/W2045045787","https://openalex.org/W2054777142","https://openalex.org/W2067111509","https://openalex.org/W2068713872","https://openalex.org/W2073707421","https://openalex.org/W2104730370","https://openalex.org/W2157283089","https://openalex.org/W2318208906","https://openalex.org/W2330283852","https://openalex.org/W2473825154","https://openalex.org/W2522608118","https://openalex.org/W2592666041","https://openalex.org/W2765175741","https://openalex.org/W2963132433","https://openalex.org/W2963221946","https://openalex.org/W2963552208","https://openalex.org/W2968995638","https://openalex.org/W2969227983","https://openalex.org/W2994764421","https://openalex.org/W3025419542"],"related_works":["https://openalex.org/W2592666041","https://openalex.org/W2891210051","https://openalex.org/W2792910721","https://openalex.org/W3105326570","https://openalex.org/W2019445751","https://openalex.org/W2563978230","https://openalex.org/W2951518706","https://openalex.org/W2439958724","https://openalex.org/W2952127590","https://openalex.org/W2004003666","https://openalex.org/W2791077531","https://openalex.org/W1973916683","https://openalex.org/W3105791322","https://openalex.org/W2004568659","https://openalex.org/W3106045873","https://openalex.org/W2765266431","https://openalex.org/W1990275182","https://openalex.org/W2768782821","https://openalex.org/W2950145919","https://openalex.org/W120315552"],"abstract_inverted_index":{"We":[0,40],"consider":[1],"a":[2,48],"fractal":[3,44],"refinement":[4],"of":[5,51],"the":[6,10,17,22,36,43,58,69,82],"Carleson":[7],"problem":[8],"for":[9,74],"Schr\u00f6dinger":[11],"equation,":[12],"that":[13],"is":[14,55],"to":[15,24,27,35,42,81],"identify":[16],"minimal":[18],"regularity":[19],"needed":[20],"by":[21],"solutions":[23],"converge":[25],"pointwise":[26,75],"their":[28],"initial":[29],"data":[30],"almost":[31],"everywhere":[32],"with":[33],"respect":[34],"$\u03b1$-Hausdorff":[37],"measure":[38,60],"($\u03b1$-a.e.).":[39],"extend":[41,79],"setting":[45,61],"($\u03b1<":[46],"n$)":[47],"recent":[49],"counterexample":[50],"Bourgain":[52],"\\cite{Bourgain2016},":[53],"which":[54],"sharp":[56],"in":[57],"Lebesque":[59],"($\u03b1=":[62],"n$).":[63],"In":[64],"doing":[65],"so":[66],"we":[67,78],"recover":[68],"necessary":[70],"condition":[71],"from":[72],"\\cite{zbMATH07036806}":[73],"convergence~$\u03b1$-a.e.":[76],"and":[77],"it":[80],"range":[83],"$n/2":[84]},"counts_by_year":[{"year":2022,"cited_by_count":2}],"updated_date":"2026-09-26T08:19:03.415552","created_date":"2025-10-10T00:00:00"}