{"id":"https://openalex.org/W7222173255","doi":"https://doi.org/10.48550/arxiv.2610.20110","title":"Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients","display_name":"Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients","publication_year":2026,"publication_date":"2026-10-14","ids":{"openalex":"https://openalex.org/W7222173255","doi":"https://doi.org/10.48550/arxiv.2610.20110"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.20110","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20110","pdf_url":"https://arxiv.org/pdf/2610.20110","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/2610.20110","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051228281","display_name":"Serena Dipierro","orcid":"https://orcid.org/0000-0003-4386-4485","observed_orcids":["https://orcid.org/0000-0003-4386-4485"]},"institutions":[{"id":"https://openalex.org/I177877127","display_name":"The University of Western Australia","ror":"https://ror.org/047272k79","country_code":"AU","type":"education","lineage":["https://openalex.org/I177877127"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Dipierro, Serena","raw_affiliation_strings":["Department of Mathematics and Statistics , The University of Western Australia , 35 Stirling Highway , Crawley , Perth , WA 6009 , Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , The University of Western Australia , 35 Stirling Highway , Crawley , Perth , WA 6009 , Australia","institution_ids":["https://openalex.org/I177877127"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028807720","display_name":"Caterina Sportelli","orcid":"https://orcid.org/0000-0002-5221-5877","observed_orcids":["https://orcid.org/0000-0002-5221-5877"]},"institutions":[{"id":"https://openalex.org/I173304897","display_name":"Universidad de Granada","ror":"https://ror.org/04njjy449","country_code":"ES","type":"education","lineage":["https://openalex.org/I173304897"]},{"id":"https://openalex.org/I177877127","display_name":"The University of Western Australia","ror":"https://ror.org/047272k79","country_code":"AU","type":"education","lineage":["https://openalex.org/I177877127"]}],"countries":["AU","ES"],"is_corresponding":false,"raw_author_name":"Sportelli, Caterina","raw_affiliation_strings":["Departamento de An\u00e1lisis Matem\u00e1tico , Universidad de Granada , 18071 Granada , Spain &","Department of Mathematics and Statistics , The University of Western Australia , 35 Stirling Highway , Crawley , Perth , WA 6009 , Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de An\u00e1lisis Matem\u00e1tico , Universidad de Granada , 18071 Granada , Spain &","institution_ids":["https://openalex.org/I173304897"]},{"raw_affiliation_string":"Department of Mathematics and Statistics , The University of Western Australia , 35 Stirling Highway , Crawley , Perth , WA 6009 , Australia","institution_ids":["https://openalex.org/I177877127"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078447699","display_name":"Enrico Valdinoci","orcid":"https://orcid.org/0000-0001-6222-2272","observed_orcids":["https://orcid.org/0000-0001-6222-2272"]},"institutions":[{"id":"https://openalex.org/I177877127","display_name":"The University of Western Australia","ror":"https://ror.org/047272k79","country_code":"AU","type":"education","lineage":["https://openalex.org/I177877127"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Valdinoci, Enrico","raw_affiliation_strings":["Department of Mathematics and Statistics , The University of Western Australia , 35 Stirling Highway , Crawley , Perth , WA 6009 , Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , The University of Western Australia , 35 Stirling Highway , Crawley , Perth , WA 6009 , Australia","institution_ids":["https://openalex.org/I177877127"]}]}],"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.6195999979972839,"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":0.6195999979972839,"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":0.14830000698566437,"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.05689999833703041,"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/fractional-gradient","display_name":"fractional gradient","score":0.5},{"id":"https://openalex.org/keywords/bilinear-forms","display_name":"bilinear forms","score":0.890999972820282},{"id":"https://openalex.org/keywords/integral-kernels","display_name":"integral kernels","score":0.625},{"id":"https://openalex.org/keywords/kernel-representation","display_name":"kernel representation","score":0.6679999828338623},{"id":"https://openalex.org/keywords/riesz-fractional-derivative","display_name":"Riesz fractional derivative","score":0.7289999723434448}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8482000231742859},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.6708999872207642},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.6690000295639038},{"id":"https://openalex.org/C152401794","wikidata":"https://www.wikidata.org/wiki/Q583760","display_name":"Antisymmetric relation","level":2,"score":0.6520000100135803},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.6503999829292297},{"id":"https://openalex.org/C79940087","wikidata":"https://www.wikidata.org/wiki/Q2100018","display_name":"Symmetric bilinear form","level":3,"score":0.5235999822616577},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.510699987411499},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45239999890327454},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.42750000953674316},{"id":"https://openalex.org/C110342517","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Constant coefficients","level":2,"score":0.40459999442100525},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.3896999955177307},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37610000371932983},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.3555000126361847},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3327000141143799},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.32690000534057617},{"id":"https://openalex.org/C44292817","wikidata":"https://www.wikidata.org/wiki/Q333871","display_name":"Orthogonal matrix","level":3,"score":0.3264000117778778},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.3239000141620636},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2973000109195709},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.27320000529289246}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.20110","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20110","pdf_url":"https://arxiv.org/pdf/2610.20110","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.2610.20110","doi":"https://doi.org/10.48550/arxiv.2610.20110","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.20110","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:2610.20110","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20110","pdf_url":"https://arxiv.org/pdf/2610.20110","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/G1287761106","display_name":null,"funder_award_id":"FL190100081","funder_id":"https://openalex.org/F4320329918","funder_display_name":"Australian Mathematical Society"},{"id":"https://openalex.org/G1403924678","display_name":null,"funder_award_id":"FL190100081","funder_id":"https://openalex.org/F4320311030","funder_display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\""},{"id":"https://openalex.org/G2199393123","display_name":null,"funder_award_id":"FT230100333","funder_id":"https://openalex.org/F4320329918","funder_display_name":"Australian Mathematical Society"}],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320329918","display_name":"Australian Mathematical Society","ror":"https://ror.org/01qmp5703"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7222173255.pdf","grobid_xml":"https://content.openalex.org/works/W7222173255.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1940830394","https://openalex.org/W2031709333","https://openalex.org/W2275926155","https://openalex.org/W2510223711","https://openalex.org/W2611549267","https://openalex.org/W2890971675","https://openalex.org/W2948339064","https://openalex.org/W3008037962","https://openalex.org/W3027140196","https://openalex.org/W3198593029","https://openalex.org/W4233350505","https://openalex.org/W4417507235"],"related_works":["https://openalex.org/W7213371149","https://openalex.org/W2075763784","https://openalex.org/W2004639026","https://openalex.org/W2032382375","https://openalex.org/W2027510516","https://openalex.org/W2076851013","https://openalex.org/W2006708518","https://openalex.org/W1604652509","https://openalex.org/W4236960255","https://openalex.org/W1966397968"],"abstract_inverted_index":{"We":[0,32,87],"study":[1],"the":[2,28,51,61,65,102,109,127],"representation":[3],"of":[4,27,50,60,92,116,130],"bilinear":[5,104],"forms":[6],"associated":[7],"with":[8],"Riesz":[9],"fractional":[10,103],"gradients":[11],"by":[12,17],"symmetric":[13,36],"integral":[14],"kernels.":[15],"Motivated":[16],"[SS18,":[18,75],"Open":[19,76],"Problem":[20,77],"1.10],":[21],"we":[22,43],"give":[23],"a":[24,57,80,90,120],"complete":[25],"description":[26],"constant":[29,52,85],"coefficient":[30],"case.":[31],"prove":[33],"that":[34,98],"any":[35],"representing":[37,110],"kernel":[38,66,111],"is":[39,67],"uniquely":[40],"determined":[41],"and":[42,106,124],"derive":[44],"its":[45],"explicit":[46],"formula":[47],"in":[48],"terms":[49],"matrix":[53],"$A$.":[54],"This":[55],"yields":[56],"sharp":[58],"characterization":[59],"matrices":[62,118],"for":[63,84],"which":[64],"either":[68],"nonnegative":[69],"or":[70],"uniformly":[71],"positive.":[72],"In":[73],"particular,":[74],"1.10]":[78],"has":[79],"negative":[81],"answer":[82],"(even":[83],"matrices).":[86],"also":[88],"identify":[89],"class":[91],"nonconstant":[93,117],"antisymmetric":[94],"divergence":[95],"free":[96],"perturbations":[97],"are":[99],"invisible":[100],"to":[101],"form":[105],"hence":[107],"leave":[108],"unchanged.":[112],"The":[113],"general":[114],"case":[115],"exhibits":[119],"substantially":[121],"different":[122],"behavior":[123],"will":[125],"be":[126],"main":[128],"subject":[129],"our":[131],"forthcoming":[132],"paper":[133],"[DSV26].":[134]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-16T00:00:00"}
