{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T11:41:18Z","timestamp":1778758878132,"version":"3.51.4"},"reference-count":28,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100006252","name":"Lule\u00e5 University of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006252","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Function Spaces and Applications"],"published-print":{"date-parts":[[2012]]},"abstract":"<jats:p>We study the weighted<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>p<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:math>-boundedness of the multidimensional weighted Hardy-type operators<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>\u210b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>with radial type weight<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>w<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>w<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">|<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">|<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, in the generalized complementary Morrey spaces<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msubsup><mml:mrow><mml:mrow><mml:mi>\u2112<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2201<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{<\/mml:mo><mml:mn mathvariant=\"normal\">0<\/mml:mn><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03c8<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>\u211d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>defined by an almost increasing function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>\u03c8<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>\u03c8<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. We prove a theorem which provides conditions, in terms of some integral inequalities imposed on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>\u03c8<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><\/mml:math>, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi>\u03c8<\/mml:mi><\/mml:mrow><\/mml:math>and the weight<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><\/mml:math>are power functions. We also prove that the spaces<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:msubsup><mml:mrow><mml:mrow><mml:mi>\u2112<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2201<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{<\/mml:mo><mml:mn mathvariant=\"normal\">0<\/mml:mn><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03c8<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u2126<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>over bounded domains \u2126 are embedded between weighted Lebesgue space<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math>with the weight<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:mi>\u03c8<\/mml:mi><\/mml:mrow><\/mml:math>and such a space with the weight<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mrow><mml:mi>\u03c8<\/mml:mi><\/mml:mrow><\/mml:math>, perturbed by a logarithmic factor. Both the embeddings are sharp.<\/jats:p>","DOI":"10.1155\/2012\/283285","type":"journal-article","created":{"date-parts":[[2012,11,11]],"date-time":"2012-11-11T16:01:07Z","timestamp":1352649667000},"page":"1-19","source":"Crossref","is-referenced-by-count":7,"title":["Weighted Hardy Operators in Complementary Morrey Spaces"],"prefix":"10.1155","volume":"2012","author":[{"given":"Dag","family":"Lukkassen","sequence":"first","affiliation":[{"name":"Department of Technology, Narvik University College, P.O. Box 385, 8505 Narvik, Norway"}]},{"given":"Lars-Erik","family":"Persson","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lule\u00e5 University of Technology, SE 921 87 Lule\u00e5, Sweden"}]},{"given":"Stefan","family":"Samko","sequence":"additional","affiliation":[{"name":"Departamento de Matematica, Universidade do Algarve, 6005-139 Faro, Portugal"}]}],"member":"311","reference":[{"key":"4","volume":"543","year":"2002"},{"key":"10","year":"2010"},{"key":"13","year":"2007"},{"key":"14","year":"2003"},{"issue":"11","key":"15","doi-asserted-by":"crossref","first-page":"1300","DOI":"10.1002\/mma.2532","volume":"35","year":"2012","journal-title":"Mathematical Methods in the Applied Sciences"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2011.09.002"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2010.11.029"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2008.09.021"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1155\/2012\/678171"},{"key":"5","year":"1983"},{"key":"12","year":"1977"},{"key":"27","series-title":"Mathematical Surveys and Monographs","volume":"81","year":"2000"},{"key":"21","volume-title":"Morrey-Campanato spaces: an overview"},{"issue":"5","key":"3","first-page":"7","year":"1982","journal-title":"Izvestiya Akademii Nauk Kazakhsko\u012d SSR. Seriya Fiziko-Matematicheskaya"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.2307\/2045731"},{"key":"7","year":"1999"},{"key":"2","series-title":"Contemporary Mathematics","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1090\/conm\/424\/08093","volume-title":"On boundedness of the fractional maximal operator from complementary Morrey-type spaces to Morrey-type spaces","volume":"424","year":"2007"},{"key":"8"},{"key":"1","first-page":"483","volume":"5","year":"1956","journal-title":"Proceedings of the Moscow Mathematical Society"},{"issue":"4","key":"9","first-page":"437","volume":"7","year":"2004","journal-title":"Fractional Calculus & Applied Analysis"},{"key":"11","year":"1978"},{"key":"16","first-page":"49","volume":"234","year":"1985","journal-title":"Dissertationes Mathematicae"},{"key":"18","doi-asserted-by":"crossref","first-page":"11","DOI":"10.4064\/sm-26-1-11-24","volume":"26","year":"1965","journal-title":"Studia Mathematica"},{"issue":"3","key":"19","first-page":"685","volume":"12","year":"2012","journal-title":"Mathematical Inequalities and Applications"},{"key":"22","first-page":"107","volume":"120","year":"1999","journal-title":"Proceedings of A. Razmadze Mathematical Institute"},{"issue":"2","key":"23","first-page":"727","volume":"30","year":"2004\/2005","journal-title":"Real Analysis Exchange"},{"key":"17","year":"1989"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2007.01.091"}],"container-title":["Journal of Function Spaces and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jfs\/2012\/283285.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jfs\/2012\/283285.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jfs\/2012\/283285.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,5]],"date-time":"2019-07-05T09:46:39Z","timestamp":1562319999000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jfs\/2012\/283285\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012]]},"references-count":28,"alternative-id":["283285","283285"],"URL":"https:\/\/doi.org\/10.1155\/2012\/283285","relation":{},"ISSN":["0972-6802","1758-4965"],"issn-type":[{"value":"0972-6802","type":"print"},{"value":"1758-4965","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012]]}}}