{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:43:30Z","timestamp":1760143410451,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,18]],"date-time":"2024-01-18T00:00:00Z","timestamp":1705536000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangzhou Huashang College Daoshi Project","award":["2023HSDS10","2022A1515012429","2021A1515010055","2021ZDJS055"],"award-info":[{"award-number":["2023HSDS10","2022A1515012429","2021A1515010055","2021ZDJS055"]}]},{"name":"Guangdong Basic and Applied Basic Research Fund","award":["2023HSDS10","2022A1515012429","2021A1515010055","2021ZDJS055"],"award-info":[{"award-number":["2023HSDS10","2022A1515012429","2021A1515010055","2021ZDJS055"]}]},{"name":"Key Construction Discipline Scientific Research Ability Promotion Project of Guangdong Province","award":["2023HSDS10","2022A1515012429","2021A1515010055","2021ZDJS055"],"award-info":[{"award-number":["2023HSDS10","2022A1515012429","2021A1515010055","2021ZDJS055"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Firstly, Hilbert-type integral inequalities with best constant factors are established for two non-homogeneous kernels. Then, by utilizing the relationship between the Hilbert-type inequality and the integral operator of same kernel, the parameter conditions for two integral operators with non-homogeneous kernels in weighted Lebesgue space to be bounded and the formula for calculating the operator norm are obtained.<\/jats:p>","DOI":"10.3390\/axioms13010058","type":"journal-article","created":{"date-parts":[[2024,1,18]],"date-time":"2024-01-18T11:28:46Z","timestamp":1705577326000},"page":"58","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Parameter Conditions for Boundedness of Two Integral Operators in Weighted Lebsgue Space and Calculation of Operator Norms"],"prefix":"10.3390","volume":"13","author":[{"given":"Lijuan","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0907-8379","authenticated-orcid":false,"given":"Bing","family":"He","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Hong","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,18]]},"reference":[{"key":"ref_1","first-page":"45","article-title":"Note on a theorem of Hilbert concerning series of positive terms","volume":"23","author":"Hardy","year":"1925","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1006\/jmaa.1997.5877","article-title":"On Hilbert\u2019s integral inequality","volume":"220","author":"Yang","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","first-page":"29","article-title":"General Hilbert\u2019s and Hardy\u2019s inequalities","volume":"8","year":"2005","journal-title":"Math. Inequal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1006\/jmaa.1999.6373","article-title":"Note on new extensions of Hilbert\u2019s integral inequality","volume":"235","author":"Kuang","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1006\/jmaa.2001.7595","article-title":"Some new type Hilbert integral inequalities","volume":"262","author":"Zhao","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1765","DOI":"10.1007\/s11785-018-0830-5","article-title":"On a Hilbert-type integral inequality in the whole plane related to the extended Riemann zeta function","volume":"13","author":"Rassias","year":"2019","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/s10476-012-0402-2","article-title":"On a multidimensional version of the Hilbert type inequality","volume":"38","year":"2012","journal-title":"Anal. Math."},{"key":"ref_8","first-page":"247","article-title":"On an extension of Hardy-Hilbert\u2019s inequality","volume":"23","author":"Yang","year":"2002","journal-title":"Ann. Math."},{"key":"ref_9","first-page":"217","article-title":"Generalization of inequalities of Hardy-Hilbert type","volume":"7","year":"2004","journal-title":"Math. Inequal. Appl."},{"key":"ref_10","first-page":"75","article-title":"On half-discrete Hilbert\u2019s inequality","volume":"220","author":"Rassias","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Yang, B.C. (2009). The Norm of Operator and Hilbert-Type Inequalities, Science Press.","DOI":"10.2174\/97816080505501090101"},{"key":"ref_12","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. (1952). Inequalities, Gambridge University Press."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1186\/s13660-019-2087-6","article-title":"A new discrete Hilbert-type inequality involving partial sums","volume":"2019","author":"Adiyasuren","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_14","first-page":"546829","article-title":"On some extensions of Hardy-Hilbert\u2019s inequality and applications","volume":"2009","author":"Azar","year":"2009","journal-title":"J. Inequal. Appl."},{"key":"ref_15","first-page":"63","article-title":"Hardy-Hilbert\u2019s inequalities with two parameters","volume":"36","author":"Xu","year":"2007","journal-title":"Adv. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1186\/s13660-015-0829-7","article-title":"A survey on the study of Hilbert-type inequalities","volume":"2015","author":"Chen","year":"2015","journal-title":"J. Inequal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"379","DOI":"10.7153\/jmi-2018-12-28","article-title":"A new form of Hilbert integral inequality","volume":"12","author":"Batbold","year":"2018","journal-title":"J. Math. Inequal."},{"key":"ref_18","first-page":"323","article-title":"The refinements of Hilbert-type inequalities in discrete case","volume":"45","year":"2018","journal-title":"Ann. Univ. Craiova-Mat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1515\/math-2021-0034","article-title":"On a new generalization of some Hilbert-type inequalities","volume":"19","author":"You","year":"2021","journal-title":"Open Math."},{"key":"ref_20","first-page":"1551","article-title":"Reverse Hilbert\u2019s type integral inequalities","volume":"17","author":"Zhao","year":"2014","journal-title":"Math. Inequal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1186\/1029-242X-2014-198","article-title":"On reverse Hilbert-type inequalities","volume":"2014","author":"Xu","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_22","first-page":"365","article-title":"On the norm of a series operator with a symmetric and homogeneous kernel and its application","volume":"51","author":"Hong","year":"2008","journal-title":"Acta Math. Sin. Chin. Ed."},{"key":"ref_23","first-page":"329","article-title":"A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel and the best constant factor","volume":"37","author":"Hong","year":"2016","journal-title":"Ann. Math."},{"key":"ref_24","first-page":"189","article-title":"Structural characteristics and applications of Hilbert\u2019s type integral inequalities with homogenous kernel","volume":"55","author":"Hong","year":"2017","journal-title":"J. Jilin Univ."},{"key":"ref_25","unstructured":"Hong, Y., and He, B. (2023). Theory and Applications of Hilbert-Type Inequalities, Science Press."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Rassias, M.T., Yang, B., and Raigorodskii, A. (2023). An equivalent form related to a Hilbert-type integral inequality. Axioms, 12.","DOI":"10.3390\/axioms12070677"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/58\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:49:58Z","timestamp":1760104198000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/58"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,18]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["axioms13010058"],"URL":"https:\/\/doi.org\/10.3390\/axioms13010058","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,1,18]]}}}