{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:47:43Z","timestamp":1775839663179,"version":"3.50.1"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"9","license":[{"start":{"date-parts":[[2002,4,17]],"date-time":"2002-04-17T00:00:00Z","timestamp":1019001600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"

\n In this paper we deal with the interpolation from Lebesgue spaces\n \n \n \n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n L^p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n L<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msup>\n L^q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , into an Orlicz space\n \n \n \n \n L<\/mml:mi>\n \n \u03c6\n \n <\/mml:mi>\n <\/mml:msup>\n L^\\varphi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n 1<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n p<\/mml:mi>\n ><\/mml:mo>\n q<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n 1\\le p>q\\le \\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n \u03c6\n \n <\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n t<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u03c1\n \n <\/mml:mi>\n (<\/mml:mo>\n \n t<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n q<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\varphi ^{-1}(t)=t^{1\/p}\\rho (t^{1\/q-1\/p})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for some concave function\n \n \n \n \n \u03c1\n \n <\/mml:mi>\n \\rho<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , with special attention to the interpolation constant\n \n \n \n C<\/mml:mi>\n C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For a bounded linear operator\n \n \n \n T<\/mml:mi>\n T<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n L^p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n L<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msup>\n L^q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,\n \n \\[\n \n \n \n \n \u2016\n \n <\/mml:mo>\n T<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n \n \u03c6\n \n <\/mml:mi>\n <\/mml:msup>\n \n \u2192\n \n <\/mml:mo>\n \n L<\/mml:mi>\n \n \u03c6\n \n <\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2264\n \n <\/mml:mo>\n C<\/mml:mi>\n max<\/mml:mo>\n \n \n {<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n \u2016\n \n <\/mml:mo>\n T<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n \n \u2192\n \n <\/mml:mo>\n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2016\n \n <\/mml:mo>\n T<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msup>\n \n \u2192\n \n <\/mml:mo>\n \n L<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\|T\\|_{L^\\varphi \\to L^\\varphi } \\le C\\max \\Big \\{ \\|T\\|_{L^p\\to L^p}, \\|T\\|_{L^q\\to L^q} \\Big \\},<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n where the interpolation constant\n \n \n \n C<\/mml:mi>\n C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n depends only on\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We give estimates for\n \n \n \n C<\/mml:mi>\n C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which imply\n \n \n \n \n C<\/mml:mi>\n ><\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n C>4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Moreover, if either\n \n \n \n \n 1<\/mml:mn>\n ><\/mml:mo>\n p<\/mml:mi>\n ><\/mml:mo>\n q<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n 1> p>q\\le 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n 2<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n p<\/mml:mi>\n ><\/mml:mo>\n q<\/mml:mi>\n ><\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n 2\\le p>q>\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then\n \n \n \n \n C<\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n C> 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . If\n \n \n \n \n q<\/mml:mi>\n =<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n q=\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then\n \n \n \n \n C<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n \n 2<\/mml:mn>\n \n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n C\\le 2^{1-1\/p}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and, in particular, for the case\n \n \n \n \n p<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n p=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n this gives the classical Orlicz interpolation theorem with the constant\n \n \n \n \n C<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n C=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0002-9939-01-06162-7","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:31Z","timestamp":1027707451000},"page":"2727-2739","source":"Crossref","is-referenced-by-count":10,"special_numbering":"507","title":["On the interpolation constant for Orlicz spaces"],"prefix":"10.1090","volume":"129","author":[{"given":"Alexei","family":"Karlovich","sequence":"first","affiliation":[]},{"given":"Lech","family":"Maligranda","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,4,17]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics","isbn-type":"print","volume-title":"Interpolation of operators","volume":"129","author":"Bennett, Colin","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0120887304"},{"key":"2","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/0022-247X(73)90193-5","article-title":"A generalization of Steffensen\u2019s inequality","volume":"41","author":"Bergh, J.","year":"1973","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"3","series-title":"Grundlehren der Mathematischen Wissenschaften, No. 223","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-66451-9","volume-title":"Interpolation spaces. An introduction","author":"Bergh, J\u00f6ran","year":"1976"},{"key":"4","doi-asserted-by":"publisher","first-page":"215","DOI":"10.2307\/2035264","article-title":"Spaces between a pair of reflexive Lebesgue spaces","volume":"18","author":"Boyd, David W.","year":"1967","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"5","series-title":"North-Holland Mathematical Library","isbn-type":"print","volume-title":"Interpolation functors and interpolation spaces. Vol. I","volume":"47","author":"Brudny\u012d, Yu. A.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0444880011"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1006\/jfan.1997.3193","article-title":"An optimal interpolation theorem of Marcinkiewicz type in Orlicz spaces","volume":"153","author":"Cianchi, Andrea","year":"1998","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"key":"7","unstructured":"I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, Pitman Monographs and Surveys in Pure and Applied Math. 92, Addison Wesley Longman 1998."},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"33","DOI":"10.4064\/sm-60-1-33-59","article-title":"Interpolation of Orlicz spaces","volume":"60","author":"Gustavsson, Jan","year":"1977","journal-title":"Studia Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0039-3223","issn-type":"print"},{"key":"9","volume-title":"Convex functions and Orlicz spaces","author":"Krasnosel\u2032ski\u012d, M. A.","year":"1961"},{"key":"10","series-title":"Translations of Mathematical Monographs","isbn-type":"print","volume-title":"Interpolation of linear operators","volume":"54","author":"Kre\u012dn, S. G.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0821845057"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"161","DOI":"10.4064\/sm-104-2-161-180","article-title":"A Carlson type inequality with blocks and interpolation","volume":"104","author":"Kruglyak, Natan Ya.","year":"1993","journal-title":"Studia Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0039-3223","issn-type":"print"},{"key":"12","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-35347-9","volume-title":"Classical Banach spaces. II","volume":"97","author":"Lindenstrauss, Joram","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/3540088881"},{"key":"13","doi-asserted-by":"publisher","first-page":"207","DOI":"10.2307\/1996007","article-title":"Interpolation theorems for the pairs of spaces (\ud835\udc3f^{\ud835\udc5d},\ud835\udc3f^{\u221e}) and (\ud835\udc3f\u00b9,\ud835\udc3f^{\ud835\udc5e})","volume":"159","author":"Lorentz, George G.","year":"1971","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"14","first-page":"49","article-title":"Indices and interpolation","volume":"234","author":"Maligranda, Lech","year":"1985","journal-title":"Dissertationes Math. (Rozprawy Mat.)","ISSN":"https:\/\/id.crossref.org\/issn\/0012-3862","issn-type":"print"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"43","DOI":"10.4064\/sm-95-1-43-58","article-title":"Some remarks on Orlicz\u2019s interpolation theorem","volume":"95","author":"Maligranda, Lech","year":"1989","journal-title":"Studia Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0039-3223","issn-type":"print"},{"key":"16","unstructured":"L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Univ. Estadual de Campinas, Campinas SP, Brazil 1989."},{"issue":"4","key":"17","first-page":"45","article-title":"Interpolation theorems that arise from Grothendieck\u2019s inequality","volume":"10","author":"Ov\u010dinnikov, V. I.","year":"1976","journal-title":"Funkcional. Anal. i Prilo\\v{z}en.","ISSN":"https:\/\/id.crossref.org\/issn\/0374-1990","issn-type":"print"},{"issue":"2","key":"18","first-page":"i--x and 349--515","article-title":"The method of orbits in interpolation theory","volume":"1","author":"Ovchinnikov, V. I.","year":"1984","journal-title":"Math. Rep.","ISSN":"https:\/\/id.crossref.org\/issn\/0275-7214","issn-type":"print"},{"key":"19","first-page":"167","article-title":"On interpolation functions","volume":"27","author":"Peetre, J.","year":"1966","journal-title":"Acta Sci. Math. (Szeged)","ISSN":"https:\/\/id.crossref.org\/issn\/0001-6969","issn-type":"print"},{"key":"20","doi-asserted-by":"publisher","first-page":"596","DOI":"10.1090\/S0002-9904-1939-07046-8","article-title":"A general continued fraction expansion","volume":"45","author":"Leighton, Walter","year":"1939","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"key":"21","first-page":"536","article-title":"Interpolation and extrapolation of linear operators in Orlicz spaces","volume":"63(105)","author":"Simonenko, I. B.","year":"1964","journal-title":"Mat. Sb. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"229","DOI":"10.4064\/sm-62-3-229-271","article-title":"Interpolation of weighted \ud835\udc3f_{\ud835\udc5d}-spaces","volume":"62","author":"Sparr, Gunnar","year":"1978","journal-title":"Studia Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0039-3223","issn-type":"print"}],"container-title":["Proceedings of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/proc\/2001-129-09\/S0002-9939-01-06162-7\/S0002-9939-01-06162-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/proc\/2001-129-09\/S0002-9939-01-06162-7\/S0002-9939-01-06162-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T15:51:26Z","timestamp":1775836286000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/proc\/2001-129-09\/S0002-9939-01-06162-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,4,17]]},"references-count":22,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2001,9]]}},"alternative-id":["S0002-9939-01-06162-7"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9939-01-06162-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6826","0002-9939"],"issn-type":[{"value":"1088-6826","type":"electronic"},{"value":"0002-9939","type":"print"}],"subject":[],"published":{"date-parts":[[2001,4,17]]}}}