{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:33:16Z","timestamp":1760146396192,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,31]],"date-time":"2024-10-31T00:00:00Z","timestamp":1730332800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Qassim University","award":["QU-APC-2024-9\/1"],"award-info":[{"award-number":["QU-APC-2024-9\/1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article contains some relations, which include some embedding and transition properties, between the Muckenhoupt classes M\u03b3;\u03b3&gt;1 and the Gehring classes G\u03b4;\u03b4&gt;1 of bi-Sobolev weights on a time scale T. In addition, we establish the relations between Muckenhoupt and Gehring classes, where we define a new time scale T\u02dc=v(T), to indicate that if the \u0394\u02dc derivative of the inverse of a bi-Sobolev weight belongs to the Gehring class, then the \u0394 derivative of a bi-Sobolev weight on a time scale T belongs to the Muckenhoupt class. Furthermore, our results, which will be established by a newly developed technique, show that the study of the properties in the continuous and discrete classes of weights can be unified. As special cases of our results, when T=R, one can obtain classical continuous results, and when T=N, one can obtain discrete results which are new and interesting for the reader.<\/jats:p>","DOI":"10.3390\/axioms13110754","type":"journal-article","created":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T13:09:27Z","timestamp":1730466567000},"page":"754","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Unified Framework for Continuous and Discrete Relations of Gehring and Muckenhoupt Weights on Time Scales"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2793-0972","authenticated-orcid":false,"given":"Samir H.","family":"Saker","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"given":"Naglaa","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6782-7908","authenticated-orcid":false,"given":"Haytham M.","family":"Rezk","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}]},{"given":"Ahmed I.","family":"Saied","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5731-3532","authenticated-orcid":false,"given":"Khaled","family":"Aldwoah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina 42351, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5898-3797","authenticated-orcid":false,"given":"Ayman","family":"Alahmade","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Taibah University, Medina 42353, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1090\/S0002-9947-1972-0293384-6","article-title":"Weighted norm inequalities for the Hardy maximal function","volume":"165","author":"Muckenhoupt","year":"1972","journal-title":"Trans. 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