A SHORT NOTE ON ENHANCED DENSITY SETS
We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at...
Saved in:
| Published in | Glasgow mathematical journal Vol. 53; no. 3; pp. 631 - 635 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.09.2011
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0017-0895 1469-509X |
| DOI | 10.1017/S001708951100022X |
Cover
| Abstract | We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana). |
|---|---|
| AbstractList | Abstract We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to 'some remarks on legendrian rectiable currents', Manuscripta Math. 113(3) (2004), 397-401) result: 'If Ω is a set of locally finite perimeter in â,,2, then there is no function f C1(â,,2) such that f(x1, x2) = (x2, 0) at a.e. (x1, x2) Ω'. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana). [PUBLICATION ABSTRACT] We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113 (3) (2004), 397–401) result: ‘ If Ω is a set of locally finite perimeter in ℝ 2 , then there is no function f ∈ C 1 (ℝ 2 ) such that ∇ f ( x 1 , x 2 ) = ( x 2 , 0) at a.e. ( x 1 , x 2 ) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L 1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C 1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana ). We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to 'some remarks on legendrian rectiable currents', Manuscripta Math. 113(3) (2004), 397-401) result: 'If Omega is a set of locally finite perimeter in 2, then there is no function f C1(2) such that f(x1, x2) = (x2, 0) at a.e. (x1, x2) Omega '. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincare-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana). We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana). |
| Author | DELLADIO, SILVANO |
| Author_xml | – sequence: 1 givenname: SILVANO surname: DELLADIO fullname: DELLADIO, SILVANO email: delladio@science.unitn.it organization: Department of Mathematics, University of Trento, via Sommarive 14, Povo, 38123 Trento, Italy e-mail: delladio@science.unitn.it |
| BookMark | eNp9kFFLwzAUhYNMcJv-AN-KIPhSvUm6pnkcW3WD0YKtMJ9CmybS0bUz6R7892ZsIEz05V4O53yXwx2hQdu1CqFbDI8YMHvKwE2I-ARjACBkfYGGOAi5PwG-HqDhwfYP_hUaWbtxkjo1RPdTL1ukr7mXpHnspYkXJ4tpMovn3jxOsmX-7mVxnl2jS100Vt2c9hi9Pcf5bOGv0pflbLryJeWk9zkDzgLCNA01YEW0jsqQVIRErKQylJhAwGQUKSBBwVgJXGNacl1VGkeBrOgYPRzv7kz3uVe2F9vaStU0Rau6vRWYBjwgFIcTF707i266vWldOxFxTjijIbgQO4ak6aw1SgtZ90Vfd21viroRGMThe-LX9xyJz8idqbeF-fqXoSem2Jamrj7UT6e_qW-c_nun |
| CitedBy_id | crossref_primary_10_1017_S0017089515000385 crossref_primary_10_1017_S0308210515000396 crossref_primary_10_1007_s00229_019_01167_0 crossref_primary_10_1017_prm_2023_22 crossref_primary_10_1017_prm_2020_85 crossref_primary_10_1007_s00229_013_0612_3 crossref_primary_10_1007_s10476_022_0178_y crossref_primary_10_1007_s00009_023_02542_8 crossref_primary_10_1007_s43037_022_00192_8 |
| Cites_doi | 10.1016/0022-1236(91)90104-D 10.1007/s00229-004-0437-1 10.1093/oso/9780198502456.001.0001 10.1515/9781400877577 10.1007/978-0-8176-4679-0 |
| ContentType | Journal Article |
| Copyright | Copyright © Glasgow Mathematical Journal Trust 2011 |
| Copyright_xml | – notice: Copyright © Glasgow Mathematical Journal Trust 2011 |
| DBID | AAYXX CITATION 3V. 7SC 7XB 88I 8AL 8FD 8FE 8FG 8FK ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BGLVJ CCPQU DWQXO GNUQQ HCIFZ JQ2 K7- L6V L7M L~C L~D M0N M2P M7S P5Z P62 PHGZM PHGZT PKEHL PQEST PQGLB PQQKQ PQUKI PRINS PTHSS Q9U |
| DOI | 10.1017/S001708951100022X |
| DatabaseName | CrossRef ProQuest Central (Corporate) Computer and Information Systems Abstracts ProQuest Central (purchase pre-March 2016) Science Database (Alumni Edition) Computing Database (Alumni Edition) Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Central Technology collection ProQuest One Community College ProQuest Central Korea ProQuest Central Student SciTech Premium Collection ProQuest Computer Science Collection Computer Science Database ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Computing Database Science Database Engineering Database Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Premium ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection ProQuest Central Basic |
| DatabaseTitle | CrossRef Computer Science Database ProQuest Central Student Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Central China ProQuest Central ProQuest One Applied & Life Sciences ProQuest Engineering Collection ProQuest Central Korea ProQuest Central (New) Advanced Technologies Database with Aerospace Engineering Collection Advanced Technologies & Aerospace Collection ProQuest Computing Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest Central Basic ProQuest Science Journals ProQuest Computing (Alumni Edition) ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest SciTech Collection Computer and Information Systems Abstracts Professional Advanced Technologies & Aerospace Database ProQuest One Academic UKI Edition Materials Science & Engineering Collection ProQuest One Academic ProQuest Central (Alumni) ProQuest One Academic (New) |
| DatabaseTitleList | Computer Science Database CrossRef Computer and Information Systems Abstracts |
| Database_xml | – sequence: 1 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1469-509X |
| EndPage | 635 |
| ExternalDocumentID | 2490612781 10_1017_S001708951100022X |
| Genre | Feature |
| GroupedDBID | --Z -1D -1F -2P -2V -E. -~6 -~N -~X .DC 09C 09E 0E1 0R~ 29I 3V. 4.4 5GY 5VS 6~7 74X 74Y 7~V 88I 8FE 8FG 8R4 8R5 9M5 AAAZR AABES AABWE AACJH AAEED AAGFV AAKTX AAMNQ AANRG AARAB AASVR AAUIS AAUKB ABBXD ABBZL ABITZ ABJCF ABJNI ABKKG ABMWE ABMYL ABQTM ABQWD ABROB ABTAH ABTCQ ABUWG ABVFV ABXAU ABZCX ABZUI ACBMC ACCHT ACETC ACGFS ACGOD ACIMK ACIWK ACMRT ACQFJ ACREK ACUIJ ACUYZ ACWGA ACYZP ACZBM ACZUX ACZWT ADCGK ADDNB ADFEC ADGEJ ADKIL ADOCW ADOVH ADOVT ADVJH AEBAK AEBPU AEHGV AEMTW AENCP AENEX AENGE AETEA AEYYC AFFUJ AFKQG AFKRA AFKSM AFLOS AFLVW AFUTZ AGABE AGBYD AGJUD AGLWM AGOOT AHQXX AHRGI AI. AIGNW AIHIV AIOIP AISIE AJ7 AJCYY AJPFC AJQAS AKZCZ ALMA_UNASSIGNED_HOLDINGS ALVPG ALWZO AQJOH ARABE ARAPS ARZZG ATUCA AUXHV AYIQA AZQEC BBLKV BCGOX BENPR BESQT BGHMG BGLVJ BJBOZ BLZWO BMAJL BPHCQ BQFHP C0O CAG CBIIA CCPQU CCQAD CCUQV CDIZJ CFAFE CFBFF CGQII CHEAL CJCSC COF CS3 DC4 DOHLZ DU5 DWQXO EBS EGQIC EJD FRP GNUQQ HCIFZ HST HZ~ H~9 I.7 I.9 IH6 IOEEP IOO IS6 I~P J36 J38 J3A JHPGK JQKCU K6V K7- KAFGG KC5 KCGVB KFECR L6V L98 LHUNA LW7 M-V M0N M2P M7S NIKVX NMFBF NZEOI O9- OHT OK1 OYBOY P2P P62 PQQKQ PROAC PTHSS PYCCK Q2X RAMDC RCA RIG ROL RR0 S6U SAAAG T9M UPT UT1 VH1 WFFJZ WH7 WQ3 WXU WXY WYP ZCG ZDLDU ZJOSE ZMEZD ZY4 ZYDXJ AAKNA AATMM AAYXX ABGDZ ABHFL ABVKB ABVZP ABXHF ACDLN ACEJA ACOZI ACRPL ADNMO ADXHL AEMFK AFZFC AGQPQ AKMAY AMVHM ANOYL CITATION PHGZM PHGZT 7SC 7XB 8AL 8FD 8FK JQ2 L7M L~C L~D PKEHL PQEST PQGLB PQUKI PRINS Q9U PUEGO |
| ID | FETCH-LOGICAL-c392t-97097427f36f01e2ff8b62d2287b3c6c12047c88e024a77b09f13b9fddf184cd3 |
| IEDL.DBID | BENPR |
| ISSN | 0017-0895 |
| IngestDate | Fri Sep 05 04:21:34 EDT 2025 Fri Jul 25 19:49:46 EDT 2025 Tue Jul 01 00:20:53 EDT 2025 Thu Apr 24 22:59:54 EDT 2025 Wed Mar 13 05:55:43 EDT 2024 |
| IsDoiOpenAccess | false |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 3 |
| Keywords | Primary 28Axx 28A75 Secondary 26A45 49Q15 |
| Language | English |
| License | https://www.cambridge.org/core/terms |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c392t-97097427f36f01e2ff8b62d2287b3c6c12047c88e024a77b09f13b9fddf184cd3 |
| Notes | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
| OpenAccessLink | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1A5EB2EA01223F8277C439A05FA136CB/S001708951100022Xa.pdf/div-class-title-a-short-note-on-enhanced-density-sets-div.pdf |
| PQID | 899297360 |
| PQPubID | 30656 |
| PageCount | 5 |
| ParticipantIDs | proquest_miscellaneous_1349423165 proquest_journals_899297360 crossref_citationtrail_10_1017_S001708951100022X crossref_primary_10_1017_S001708951100022X cambridge_journals_10_1017_S001708951100022X |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2011-09-01 |
| PublicationDateYYYYMMDD | 2011-09-01 |
| PublicationDate_xml | – month: 09 year: 2011 text: 2011-09-01 day: 01 |
| PublicationDecade | 2010 |
| PublicationPlace | Cambridge, UK |
| PublicationPlace_xml | – name: Cambridge, UK – name: Cambridge |
| PublicationTitle | Glasgow mathematical journal |
| PublicationTitleAlternate | Glasgow Math. J |
| PublicationYear | 2011 |
| Publisher | Cambridge University Press |
| Publisher_xml | – name: Cambridge University Press |
| References | S001708951100022X_ref3 S001708951100022X_ref4 S001708951100022X_ref1 Ambrosio (S001708951100022X_ref2) 2000 S001708951100022X_ref5 S001708951100022X_ref6 |
| References_xml | – ident: S001708951100022X_ref1 doi: 10.1016/0022-1236(91)90104-D – ident: S001708951100022X_ref4 doi: 10.1007/s00229-004-0437-1 – volume-title: Functions of bounded variation and free discontinuity problems year: 2000 ident: S001708951100022X_ref2 doi: 10.1093/oso/9780198502456.001.0001 – ident: S001708951100022X_ref6 doi: 10.1515/9781400877577 – ident: S001708951100022X_ref3 – ident: S001708951100022X_ref5 doi: 10.1007/978-0-8176-4679-0 |
| SSID | ssj0013089 |
| Score | 1.9147342 |
| Snippet | We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004),... We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113 (3) (2004),... Abstract We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to 'some remarks on legendrian rectiable currents', Manuscripta Math. 113(3)... We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to 'some remarks on legendrian rectiable currents', Manuscripta Math. 113(3) (2004),... |
| SourceID | proquest crossref cambridge |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 631 |
| SubjectTerms | Approximation Density Mathematical analysis Mathematics Proving |
| Title | A SHORT NOTE ON ENHANCED DENSITY SETS |
| URI | https://www.cambridge.org/core/product/identifier/S001708951100022X/type/journal_article https://www.proquest.com/docview/899297360 https://www.proquest.com/docview/1349423165 |
| Volume | 53 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1JT4NAFH7pctGDcY212mCiFyNxOsAAB2OqpaJJqSk0qaeG2U6GVtv-f2daoDYmvRFgyPBmeDvfB3CDlRPCEUlNZTuwaQvfMqlI2yZ3hCSOELZNdR6yH5FwZL-PnXEF-sW_MLqtstCJK0XNp0znyB9UXKBplgh6mn2bmjRKF1cLBo00Z1bgjyuEsSrUsSZVrkH9OYg-hpuyAvL8QjWrQ6coc64wpDWSjDqnMdSUYRv_BVvYNlrbOntliHqHcJB7kEZnveRHUBHZMez3S_jV-Qncdow4HAwTIxokgTGIjCAKNQlN1-gGUfyWfBpxkMSnMOoFyUto5nwIJlNezML0XaS8f-xKi0jUFlhKjxLMsQp6qMUIa2Nku8zzhLK7qetS5Mu2RX3JuVRxHOPWGdSyaSbOwcDIoSllXqpXIyUOlRQ5OkLljBHq-w24L19-ku_q-WTdEeZO_smqAaiQz4Tl2OKa4uJr15C7cshsDayx6-ZmIfTNbMod0YDr8qr6OHTFI83EdKkmrMF3lAdLnIudT2jC3jpXrHvHLqG2-FmKK-VsLGgLql7vtZVvpF_Bwsk_ |
| linkProvider | ProQuest |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3PT9swFH5icGAc0AabKDAwElzQorlO4iSHamI0KAWaIhqk7pTFv06oLbQV4o_b_7bnNgmgSb1xi5LYsp-f_b3nZ38P4JihEaIoLxzEDuZ4OnIdoYumo3xtuK-15wm7D9lNeXLnXQ78wQr8re7C2GOV1Zo4X6jVSNo98h_oF9g0S5z-HD84NmmUDa5WGTSKMrOCas0Zxsp7HVf6-Qk9uEmr08bhPmHsIs7OE6dMMuBINA2mThRQNKlZYFxuaFMzY0LBmWLoSQhXctlk1AtkGGoEsyIIBI1M0xWRUcqgcySVi_V-gDW0OlycVGu_4vTm9iWMQcOoggJ89Kuw6pyz2jLX4DvL2YZAOnhN7vAWJN9ixBz4Lj7BZmmxkrOFin2GFT3cgo1uTfc62YaTM9JPercZSXtZTHopidPEJr1pk3ac9jvZb9KPs_4XuHsX0XyF1eFoqHeAMOqLQsiwsKNfcF8YQX3rESspuYiiBnyvO5-Xs2iSL06gBfl_smoAreSTy5LL3KbUuF9W5LQuMl4QeSz7ea8S-ktrag1swFH9FSejjbAUQz2aYYMt2Q9azNzfXVrDIawnWfc6v-6kV3vwcbFPbc-t7cPq9HGmv6GhMxUHpToR-PPeGvwPRpcEdw |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+SHORT+NOTE+ON+ENHANCED+DENSITY+SETS&rft.jtitle=Glasgow+mathematical+journal&rft.au=DELLADIO%2C+SILVANO&rft.date=2011-09-01&rft.pub=Cambridge+University+Press&rft.issn=0017-0895&rft.eissn=1469-509X&rft.volume=53&rft.issue=3&rft.spage=631&rft_id=info:doi/10.1017%2FS001708951100022X&rft.externalDocID=2490612781 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0017-0895&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0017-0895&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0017-0895&client=summon |