{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T06:58:27Z","timestamp":1787381907236,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 1283\/2 2021 - 317210226"],"award-info":[{"award-number":["SFB 1283\/2 2021 - 317210226"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2111228"],"award-info":[{"award-number":["DMS-2111228"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2024,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Let [Formula: see text] be a convex polytope ([Formula: see text]). The Ritz projection is the best approximation, in the [Formula: see text]-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, the gradient at any point in [Formula: see text] is controlled by the Hardy\u2013Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of interest in the analysis of PDEs immediately follows. Among those are weighted spaces, Orlicz spaces, and Lorentz spaces.<\/jats:p>","DOI":"10.1137\/23m1571800","type":"journal-article","created":{"date-parts":[[2024,5,21]],"date-time":"2024-05-21T04:02:21Z","timestamp":1716264141000},"page":"1212-1225","source":"Crossref","is-referenced-by-count":4,"title":["Pointwise Gradient Estimate of the Ritz Projection"],"prefix":"10.1137","volume":"62","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0523-3079","authenticated-orcid":true,"given":"Lars","family":"Diening","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-3148-3388","authenticated-orcid":true,"given":"Julian","family":"Rolfes","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abner J.","family":"Salgado","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tennessee, Knoxville, TN 37996 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,5,21]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.2307\/2006999"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66451-9"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.2478\/s13540-011-0023-7"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/7155"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0548-3"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2011-02546-9"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1515\/FORUM.2008.027"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18363-8"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/18M1213105"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1993-1156467-3"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-1194-3"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-1230-8"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01166225"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-009-0213-y"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-01-06162-7"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1142\/1367"},{"key":"ref17","volume-title":"Convex Functions and Orlicz Spaces","author":"Krasnoselskii M. A.","year":"1961"},{"key":"ref17a","doi-asserted-by":"publisher","DOI":"10.1137\/15M1013912"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01419529"},{"key":"ref19","first-page":"18","volume-title":"Journ\u00e9es \u201c\u00c9l\u00e9ments Finis\u201d (Rennes, 1975)","author":"Nitsche J. A.","year":"1975"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-014-9208-x"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1982-0645661-4"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1990-1011446-7"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1007\/BF01436386"},{"key":"ref23","first-page":"536","volume":"63","author":"Simonenko I. B.","year":"1964","journal-title":"Mat. Sb. (N.S.)"},{"key":"ref24","series-title":"Princeton Math. Ser. 30","volume-title":"Singular Integrals and Differentiability Properties of Functions","author":"Stein E. M.","year":"1970"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1007\/s002110000184"},{"key":"ref25","unstructured":"A. Torchinsky, Real-Variable Methods in Harmonic Analysis, reprint of the 1986 original, Dover, Mineola, NY, 2004."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M1571800","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:41:47Z","timestamp":1787326907000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M1571800"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,21]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,6,30]]}},"alternative-id":["10.1137\/23M1571800"],"URL":"https:\/\/doi.org\/10.1137\/23m1571800","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,21]]}}}