{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:56:45Z","timestamp":1760147805382,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,3]],"date-time":"2023-03-03T00:00:00Z","timestamp":1677801600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Tianjin City, China","award":["18JCYBJC16300","451-03-68\/2020\/14\/200156","MOST 111-2115-M-017-002"],"award-info":[{"award-number":["18JCYBJC16300","451-03-68\/2020\/14\/200156","MOST 111-2115-M-017-002"]}]},{"name":"Ministry of Science and Technological Development, Republic of Serbia","award":["18JCYBJC16300","451-03-68\/2020\/14\/200156","MOST 111-2115-M-017-002"],"award-info":[{"award-number":["18JCYBJC16300","451-03-68\/2020\/14\/200156","MOST 111-2115-M-017-002"]}]},{"name":"National Science and Technology Council of the Republic of China","award":["18JCYBJC16300","451-03-68\/2020\/14\/200156","MOST 111-2115-M-017-002"],"award-info":[{"award-number":["18JCYBJC16300","451-03-68\/2020\/14\/200156","MOST 111-2115-M-017-002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we introduce and study the concept of the ingredient modulus of smoothness in component form in Lp\u2192(Rd) and a kind of mixed-norm Sobolev space. We obtain some new properties, inequalities, and auxiliary results in mixed-norm spaces Lp\u2192(Rd). In addition, a new concept of mixed-norm Besov space is presented and a new decomposition theorem for mixed-norm Besov spaces is established.<\/jats:p>","DOI":"10.3390\/sym15030642","type":"journal-article","created":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T05:18:43Z","timestamp":1678079923000},"page":"642","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On New Decomposition Theorems for Mixed-Norm Besov Spaces with Ingredient Modulus of Smoothness"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5222-0994","authenticated-orcid":false,"given":"Junjian","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Tiangong University, Tianjin 300387, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0392-4976","authenticated-orcid":false,"given":"Marko","family":"Kosti\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Novi Sad, Trg D. 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