{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:00:54Z","timestamp":1760058054134,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T00:00:00Z","timestamp":1741824000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper focuses on studying the generalized Marcinkiewicz integral operators with mixed homogeneity over symmetric spaces. By making an appropriate decomposition of the aforementioned operators and tracking certain estimates, the boundedness of these operators is established from the homogeneous Triebel\u2013Lizorkin space F.p0,\u03c4(Rd) to the Lp(Rd) space for all p,\u03c4\u2208(2+2\u03b11+2\u03b1,2\u03b1+2) provided that the kernel functions belong to the Grafakos\u2013Stefanov class. The main results generalize and improve some previously known results on Marcinkiewicz and generalized Marcinkiewicz operators.<\/jats:p>","DOI":"10.3390\/sym17030429","type":"journal-article","created":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T06:53:00Z","timestamp":1741848780000},"page":"429","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on Generalized Parabolic Marcinkiewicz Integrals with Grafakos\u2013Stefanov Kernels"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0188-682X","authenticated-orcid":false,"given":"Hussain","family":"Al-Qassem","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Qatar University, Doha P.O. Box 2713, Qatar"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3381-9234","authenticated-orcid":false,"given":"Mohammed","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.4064\/sm-27-1-19-38","article-title":"Singular integrals with mixed homogeneity","volume":"27","author":"Fabes","year":"1966","journal-title":"Studia Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1090\/S0002-9947-1958-0112932-2","article-title":"On the functions of Littlewood-Paley, Lusin and Marcinkiewicz","volume":"88","author":"Stein","year":"1958","journal-title":"Trans. Amer. Math. Soc."},{"unstructured":"Stein, E. (1986, January 3\u201311). Problems in harmonic analysis related to curvature and oscillatory integrals. Proceedings of the International Congress of Mathematicians, Berkeley, CA, USA.","key":"ref_3"},{"key":"ref_4","first-page":"434","article-title":"Some geometrical concepts arising in harmonic analysis","volume":"1","author":"Stein","year":"2011","journal-title":"Geom. Funct. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"203","DOI":"10.4064\/sm-44-3-203-217","article-title":"On the function of Marcinkiewicz","volume":"44","author":"Walsh","year":"1972","journal-title":"Studia Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"697","DOI":"10.4310\/MRL.2002.v9.n5.a11","article-title":"Lp bounds for the function of Marcinkiewicz","volume":"9","author":"Cheng","year":"2002","journal-title":"Math. Res. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1016\/S0022-247X(03)00244-0","article-title":"A note on Marcinkiewicz integral operators","volume":"282","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1002\/1522-2616(200107)227:1<33::AID-MANA33>3.0.CO;2-0","article-title":"A note on a Marcinkiewicz integral operator","volume":"227","author":"Chen","year":"2001","journal-title":"Math. Nachr."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF02547187","article-title":"Estimates for translation invariant operators in Lp space","volume":"104","year":"1960","journal-title":"Acta Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1017\/S1446788700003542","article-title":"A problem on rough parametric Marcinkiewicz functions","volume":"72","author":"Ding","year":"2002","journal-title":"J. Aust. Math. Soc."},{"key":"ref_11","first-page":"103","article-title":"Boundedness of Marcinkiewicz functions","volume":"135","author":"Sakamota","year":"1999","journal-title":"Studia Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"235","DOI":"10.4064\/cm-60-61-1-235-243","article-title":"A note on the Marcinkiewicz integral","volume":"61","author":"Torchinsky","year":"1990","journal-title":"Colloq. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1186\/1029-242X-2014-269","article-title":"Lp Estimates for Marcinkiewicz Integral Operators and Extrapolation","volume":"1","author":"Ali","year":"2014","journal-title":"J. Ineq. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s11464-019-0742-3","article-title":"Boundedness and continuity of Marcinkiewicz integrals associated to homogeneous mappings on Triebel-Lizorkin spaces","volume":"14","author":"Liu","year":"2019","journal-title":"Front. Math. China"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2049","DOI":"10.1007\/s10114-008-6338-6","article-title":"Parabolic Littlewood-Paley g-function with rough kernels","volume":"24","author":"Xue","year":"2008","journal-title":"Acta Math. Sin. (Engl. Ser.)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"733","DOI":"10.4134\/JKMS.2007.44.3.733","article-title":"Lp Bounds for the parabolic Marcinkiewicz integral with rough kernels","volume":"44","author":"Chen","year":"2007","journal-title":"J. Korean Math. Soc."},{"key":"ref_17","first-page":"21","article-title":"A note on parabolic Marcinkiewicz integrals along surfaces","volume":"154","year":"2010","journal-title":"Trans. A Razmadze Math. Inst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"787","DOI":"10.4134\/BKMS.2012.49.4.787","article-title":"Lp bounds for the parabolic Littlewood-Paley operator associated to surfaces of revolution","volume":"29","author":"Wang","year":"2012","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1017\/S1446788717000143","article-title":"A note on Marcinkiewicz integral associated to surfaces of revolution","volume":"104","author":"Liu","year":"2018","journal-title":"J. Aust. Math. Soc."},{"key":"ref_20","first-page":"44","article-title":"Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels","volume":"53","author":"Ali","year":"2020","journal-title":"Demon. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1017\/S1446788715000191","article-title":"Lp bounds for nonisotropic Marcinkiewicz integrals associated to surfaces","volume":"99","author":"Liu","year":"2015","journal-title":"J. Aust. Math. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s43034-023-00281-1","article-title":"On weighted boundedness and compactness of commutators of Marcinkiewicz integral associated with Schr\u00f6dinger operators","volume":"14","author":"Zhang","year":"2023","journal-title":"Ann. Func. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12044-019-0516-6","article-title":"BLO estimates for Marcinkiewicz integrals associated with Schr\u00f6dinger operators","volume":"129","author":"Gao","year":"2019","journal-title":"Proc. Indian Acad. Sci. Math. Sci."},{"key":"ref_24","first-page":"53","article-title":"A note concerning Marcinkiewicz integral with rough kernel","volume":"24","year":"2021","journal-title":"Infin. Dimens. Anal. Quantum Probab. Relat. Top."},{"key":"ref_25","first-page":"145","article-title":"Boundedness of Marcinkiewicz integral with rough kernel and their commutator on weighted Herz space with variable exponent","volume":"2","author":"Khalil","year":"2023","journal-title":"Acta Math. Univ. Comen."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00025-022-01805-2","article-title":"Estimates for Littlewood\u2013Paley operators on ball Campanato-type function spaces","volume":"78","author":"Jia","year":"2023","journal-title":"Results Math."},{"key":"ref_27","first-page":"1","article-title":"Invariant means and lacunary sequencespaces of order (\u03b1, \u03b2)","volume":"57","author":"Nasiruzzaman","year":"2024","journal-title":"Demonstr. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1007\/s40314-024-02946-6","article-title":"A note on a general sequence of \u03bb-Sz\u00e1sz Kantorovich type operators","volume":"43","author":"Rao","year":"2024","journal-title":"Comput. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4409","DOI":"10.3934\/math.2024217","article-title":"Approximation by the modified \u03bb-Bernstein-polynomial in terms of basis function","volume":"9","author":"Nasiruzzaman","year":"2024","journal-title":"AIMS Math."},{"key":"ref_30","first-page":"3261","article-title":"On q-statistical approximation of wavelets aided Kantorovich q-Baskakov operators","volume":"38","author":"Lamichhane","year":"2024","journal-title":"FILOMAT"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/S0022-247X(02)00419-5","article-title":"Singular integral operators on function spaces","volume":"276","author":"Chen","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1016\/j.jmaa.2008.05.018","article-title":"Singular integrals with mixed homogeneity in Triebel-Lizorkin spaces","volume":"345","author":"Le","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_33","first-page":"763","article-title":"On rough generalized parametric Marcinkiewicz integrals","volume":"11","author":"Cheng","year":"2017","journal-title":"J. Math. Ineq."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"261","DOI":"10.5565\/PUBLMAT_56212_01","article-title":"Boundedness of rough integral operators on Triebel-Lizorkin space","volume":"56","author":"Cheng","year":"2012","journal-title":"Publ. Mat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"100","DOI":"10.4153\/CMB-2010-085-3","article-title":"On the generalized Marcinkiewicz integral operators with rough kernels","volume":"54","author":"Fan","year":"2011","journal-title":"Can. Math. Bull."},{"key":"ref_36","first-page":"1900","article-title":"Boundedness of a class of rough maximal functions","volume":"305","author":"Ali","year":"2018","journal-title":"J. Ineq. Appl."},{"key":"ref_37","first-page":"1099","article-title":"A note on generalized parametric Marcinkiewicz integrals","volume":"56","author":"Liu","year":"2019","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1515\/math-2022-0603","article-title":"Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds","volume":"21","author":"Ali","year":"2023","journal-title":"Open Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1512\/iumj.1998.47.1521","article-title":"Lp bounds for singular integrals and maximal singular integrals with rough kernel","volume":"47","author":"Grafakos","year":"1998","journal-title":"Indiana Univ. Math. J."},{"doi-asserted-by":"crossref","unstructured":"Triebel, H. (1983). Theory of Function Spaces, Birkh\u00e4user. [1st ed.].","key":"ref_40","DOI":"10.1007\/978-3-0346-0416-1"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/0022-1236(87)90064-4","article-title":"Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals","volume":"73","author":"Ricci","year":"1987","journal-title":"J. Funct. Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/429\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:52:56Z","timestamp":1760028776000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/429"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,13]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["sym17030429"],"URL":"https:\/\/doi.org\/10.3390\/sym17030429","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,3,13]]}}}