Ontology type: schema:ScholarlyArticle Open Access: True
2016-02
AUTHORSGiovanni Alberti, Andrea Marchese
ABSTRACTFor every finite measure μ on Rn we define a decomposability bundle V(μ,·) related to the decompositions of μ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on Rn is differentiable at μ-a.e. x with respect to the subspace V(μ,x), and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at μ-a.e. x in any direction which is not in V(μ,x). As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) k-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class C1 to Lipschitz maps. More... »
PAGES1-66
http://scigraph.springernature.com/pub.10.1007/s00039-016-0354-y
DOIhttp://dx.doi.org/10.1007/s00039-016-0354-y
DIMENSIONShttps://app.dimensions.ai/details/publication/pub.1035724589
JSON-LD is the canonical representation for SciGraph data.
TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT
[
{
"@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json",
"about": [
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0601",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Biochemistry and Cell Biology",
"type": "DefinedTerm"
},
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/06",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Biological Sciences",
"type": "DefinedTerm"
}
],
"author": [
{
"affiliation": {
"alternateName": "University of Pisa",
"id": "https://www.grid.ac/institutes/grid.5395.a",
"name": [
"Dipartimento di Matematica, Universit\u00e0 di Pisa, largo Pontecorvo 5, 56127, Pisa, Italy"
],
"type": "Organization"
},
"familyName": "Alberti",
"givenName": "Giovanni",
"id": "sg:person.016436042355.89",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016436042355.89"
],
"type": "Person"
},
{
"affiliation": {
"alternateName": "University of Zurich",
"id": "https://www.grid.ac/institutes/grid.7400.3",
"name": [
"Institut f\u00fcr Mathematik, Universit\u00e4t Z\u00fcrich, Winterthurerstrasse 190, 8057, Zurich, Switzerland"
],
"type": "Organization"
},
"familyName": "Marchese",
"givenName": "Andrea",
"id": "sg:person.07377045013.62",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07377045013.62"
],
"type": "Person"
}
],
"citation": [
{
"id": "https://doi.org/10.1016/0022-1236(90)90147-d",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1022549597"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/s11856-012-0014-3",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1023864857",
"https://doi.org/10.1007/s11856-012-0014-3"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1090/s0894-0347-2014-00810-9",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1023919506"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1016/j.jfa.2012.12.007",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1024072231"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/s000390050094",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1025720597",
"https://doi.org/10.1007/s000390050094"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1016/s0764-4442(00)01792-4",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1030376165"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-3-642-85473-6",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1032502219",
"https://doi.org/10.1007/978-3-642-85473-6"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-3-642-85473-6",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1032502219",
"https://doi.org/10.1007/978-3-642-85473-6"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1112/s0024609302001157",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1036565990"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/s00222-014-0520-5",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1040345921",
"https://doi.org/10.1007/s00222-014-0520-5"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-1-4612-4190-4",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1040723300",
"https://doi.org/10.1007/978-1-4612-4190-4"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-1-4612-4190-4",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1040723300",
"https://doi.org/10.1007/978-1-4612-4190-4"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1016/s0001-8708(03)00089-6",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1041778555"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1016/s0001-8708(03)00089-6",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1041778555"
],
"type": "CreativeWork"
},
{
"id": "https://app.dimensions.ai/details/publication/pub.1050689782",
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-0-8176-4679-0",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1050689782",
"https://doi.org/10.1007/978-0-8176-4679-0"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-0-8176-4679-0",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1050689782",
"https://doi.org/10.1007/978-0-8176-4679-0"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1017/s030821050002566x",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1054894249"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1215/s0012-7094-69-03695-3",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1064418250"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.24033/bsmf.1381",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1083660311"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.4171/009-1/1",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1087826690"
],
"type": "CreativeWork"
},
{
"id": "https://doi.org/10.1515/9781400842698",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1097023077"
],
"type": "CreativeWork"
}
],
"datePublished": "2016-02",
"datePublishedReg": "2016-02-01",
"description": "For every finite measure \u03bc on Rn we define a decomposability bundle V(\u03bc,\u00b7) related to the decompositions of \u03bc in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on Rn is differentiable at \u03bc-a.e. x with respect to the subspace V(\u03bc,x), and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at \u03bc-a.e. x in any direction which is not in V(\u03bc,x). As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) k-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class C1 to Lipschitz maps.",
"genre": "research_article",
"id": "sg:pub.10.1007/s00039-016-0354-y",
"inLanguage": [
"en"
],
"isAccessibleForFree": true,
"isPartOf": [
{
"id": "sg:journal.1136023",
"issn": [
"1016-443X",
"1420-8970"
],
"name": "Geometric and Functional Analysis",
"type": "Periodical"
},
{
"issueNumber": "1",
"type": "PublicationIssue"
},
{
"type": "PublicationVolume",
"volumeNumber": "26"
}
],
"name": "On the differentiability of Lipschitz functions with respect to measures in the Euclidean space",
"pagination": "1-66",
"productId": [
{
"name": "readcube_id",
"type": "PropertyValue",
"value": [
"aa2dae46f873159c8b1711ade2dbfbdfbae6009e67f62b634c8b7d061931bb29"
]
},
{
"name": "doi",
"type": "PropertyValue",
"value": [
"10.1007/s00039-016-0354-y"
]
},
{
"name": "dimensions_id",
"type": "PropertyValue",
"value": [
"pub.1035724589"
]
}
],
"sameAs": [
"https://doi.org/10.1007/s00039-016-0354-y",
"https://app.dimensions.ai/details/publication/pub.1035724589"
],
"sdDataset": "articles",
"sdDatePublished": "2019-04-10T14:58",
"sdLicense": "https://scigraph.springernature.com/explorer/license/",
"sdPublisher": {
"name": "Springer Nature - SN SciGraph project",
"type": "Organization"
},
"sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8663_00000500.jsonl",
"type": "ScholarlyArticle",
"url": "http://link.springer.com/10.1007/s00039-016-0354-y"
}
]Download the RDF metadata as: json-ld nt turtle xml License info
JSON-LD is a popular format for linked data which is fully compatible with JSON.
curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00039-016-0354-y'
N-Triples is a line-based linked data format ideal for batch operations.
curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00039-016-0354-y'
Turtle is a human-readable linked data format.
curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00039-016-0354-y'
RDF/XML is a standard XML format for linked data.
curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00039-016-0354-y'
This table displays all metadata directly associated to this object as RDF triples.
130 TRIPLES
21 PREDICATES
45 URIs
19 LITERALS
7 BLANK NODES