{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T14:47:15Z","timestamp":1777301235150,"version":"3.51.4"},"reference-count":0,"publisher":"Wiley","issue":"57","license":[{"start":{"date-parts":[[2003,10,11]],"date-time":"2003-10-11T00:00:00Z","timestamp":1065830400000},"content-version":"vor","delay-in-days":283,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>A Volterra integral equation of the first kind  with a locally\nintegrable kernel  is called\nSonine equation if there exists another locally integrable\nkernel  <jats:italic>\u2113<\/jats:italic>(<jats:italic>x<\/jats:italic>) such that  (locally integrable divisors of the unit, with respect to the\noperation of convolution). The formal inversion\n is well known, but\nit does not work, for example, on solutions in the spaces\n<jats:italic>X<\/jats:italic> = <jats:italic>L<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>(<jats:italic>\u211d<\/jats:italic><jats:sup>1<\/jats:sup>) and is not defined on the whole range\n<jats:italic>K<\/jats:italic>(<jats:italic>X<\/jats:italic>). We develop many properties of Sonine kernels which allow\nus\u2014in a very general case\u2014to construct the real inverse\noperator, within the framework of the spaces <jats:italic>L<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>(<jats:italic>\u211d<\/jats:italic><jats:sup>1<\/jats:sup>),\nin Marchaud form:  with the interpretation of the\nconvergence of this hypersingular integral in <jats:italic>L<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>\u2010norm. The\ndescription of the range <jats:italic>K<\/jats:italic>(<jats:italic>X<\/jats:italic>) is given; it already requires the\nlanguage of Orlicz spaces even in the case when <jats:italic>X<\/jats:italic> is the\nLebesgue space <jats:italic>L<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>(<jats:italic>\u211d<\/jats:italic><jats:sup>1<\/jats:sup>).<\/jats:p>","DOI":"10.1155\/s0161171203211455","type":"journal-article","created":{"date-parts":[[2003,10,13]],"date-time":"2003-10-13T09:47:01Z","timestamp":1066038421000},"page":"3609-3632","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":93,"title":["Integral equations of the first kind of Sonine type"],"prefix":"10.1155","volume":"2003","author":[{"given":"Stefan G.","family":"Samko","sequence":"first","affiliation":[]},{"given":"Rog\u00e9rio P.","family":"Cardoso","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2003,10,11]]},"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2003\/238394.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/S0161171203211455","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T20:00:58Z","timestamp":1719518458000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/S0161171203211455"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":0,"journal-issue":{"issue":"57","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1155\/S0161171203211455"],"URL":"https:\/\/doi.org\/10.1155\/s0161171203211455","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]},"assertion":[{"value":"2002-11-27","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2003-10-11","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}