{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,9]],"date-time":"2025-11-09T17:45:43Z","timestamp":1762710343288,"version":"3.40.5"},"reference-count":18,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,6,1]]},"abstract":"Abstract<\/jats:title>\n We show that if the Hardy\u2013Littewood maximal operator M<\/jats:italic> is bounded on a\nreflexive variable exponent space \n \n \n \n \n L<\/m:mi>\n \n p<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u22c5<\/m:mo>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \u211d<\/m:mi>\n d<\/m:mi>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {L^{p(\\,\\cdot\\,)}(\\mathbb{R}^{d})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, then for\nevery \n \n \n \n q<\/m:mi>\n \u2208<\/m:mo>\n \n (<\/m:mo>\n 1<\/m:mn>\n ,<\/m:mo>\n \u221e<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {q\\in(1,\\infty)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the exponent \n \n \n \n p<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u22c5<\/m:mo>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {p(\\,\\cdot\\,)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> admits, for all sufficiently\nsmall \n \n \n \n \u03b8<\/m:mi>\n ><\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {\\theta>0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the representation \n \n \n \n \n 1<\/m:mn>\n \n p<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mfrac>\n =<\/m:mo>\n \n \n \u03b8<\/m:mi>\n q<\/m:mi>\n <\/m:mfrac>\n +<\/m:mo>\n \n \n 1<\/m:mn>\n -<\/m:mo>\n \u03b8<\/m:mi>\n <\/m:mrow>\n \n r<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mfrac>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\frac{1}{p(x)}=\\frac{\\theta}{q}+\\frac{1-\\theta}{r(x)}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>,\n\n \n \n \n x<\/m:mi>\n \u2208<\/m:mo>\n \n \u211d<\/m:mi>\n d<\/m:mi>\n <\/m:msup>\n <\/m:mrow>\n <\/m:math>\n \n {x\\in\\mathbb{R}^{d}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, such that the operator M<\/jats:italic> is bounded on the variable\nLebesgue space \n \n \n \n \n L<\/m:mi>\n \n r<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u22c5<\/m:mo>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \u211d<\/m:mi>\n d<\/m:mi>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {L^{r(\\,\\cdot\\,)}(\\mathbb{R}^{d})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThis result can be applied for transferring properties like\ncompactness of linear operators from standard Lebesgue spaces to\nvariable Lebesgue spaces by using interpolation techniques.<\/jats:p>","DOI":"10.1515\/gmj-2022-2152","type":"journal-article","created":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T19:17:40Z","timestamp":1648235860000},"page":"347-352","source":"Crossref","is-referenced-by-count":3,"title":["On interpolation of reflexive variable Lebesgue spaces on which the Hardy\u2013Littlewood maximal operator is bounded"],"prefix":"10.1515","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0523-3079","authenticated-orcid":false,"given":"Lars","family":"Diening","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Bielefeld , Postfach 10 01 31, 33501 Bielefeld , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6815-0561","authenticated-orcid":false,"given":"Oleksiy","family":"Karlovych","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica , Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es, Faculdade de Ci\u00eancias e Tecnologia , Universidade Nova de Lisboa , Quinta da Torre, 2829\u2013516 Caparica , Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9145-8978","authenticated-orcid":false,"given":"Eugene","family":"Shargorodsky","sequence":"additional","affiliation":[{"name":"Department of Mathematics , King\u2019s College London , Strand , London WC2R 2LS , United Kingdom ; and Technische Universit\u00e4t Dresden, Fakult\u00e4t Mathematik, 01062 Dresden, Germany"}]}],"member":"374","published-online":{"date-parts":[[2022,3,26]]},"reference":[{"key":"2023033119260594563_j_gmj-2022-2152_ref_001","doi-asserted-by":"crossref","unstructured":"D. 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Math. 129 (2005), no. 8, 657\u2013700.","DOI":"10.1016\/j.bulsci.2003.10.003"},{"key":"2023033119260594563_j_gmj-2022-2152_ref_005","doi-asserted-by":"crossref","unstructured":"L. Diening, P. Harjulehto, P. H\u00e4st\u00f6 and M. R\u016f\u017ei\u010dka,\nLebesgue and Sobolev Spaces with Variable Exponents,\nLecture Notes in Math. 2017,\nSpringer, Heidelberg, 2011.","DOI":"10.1007\/978-3-642-18363-8"},{"key":"2023033119260594563_j_gmj-2022-2152_ref_006","unstructured":"L. Diening, P. H\u00e4st\u00f6 and A. Nekvinda,\nOpen problems in variable Lebesgue and Sobolev spaces,\nFSDONA04 Proceedings,\nCzech Academy of Sciences, Milovy (2004), 38\u201358."},{"key":"#cr-split#-2023033119260594563_j_gmj-2022-2152_ref_007.1","doi-asserted-by":"crossref","unstructured":"A. Fiorenza, A. Gogatishvili and T. Kopaliani, Estimates for imaginary powers of the Laplace operator in variable Lebesgue spaces and applications, Izv. Nats. Akad. 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Math. 140 (2016), no. 4, 86\u201397.","DOI":"10.1016\/j.bulsci.2015.04.003"},{"key":"2023033119260594563_j_gmj-2022-2152_ref_010","doi-asserted-by":"crossref","unstructured":"A. Y. Karlovich,\nAlgebras of continuous Fourier multipliers on variable Lebesgue spaces,\nMediterr. J. Math. 17 (2020), no. 4, Paper No. 102.","DOI":"10.1007\/s00009-020-01537-z"},{"key":"2023033119260594563_j_gmj-2022-2152_ref_011","doi-asserted-by":"crossref","unstructured":"A. Y. Karlovich and I. M. Spitkovsky,\nPseudodifferential operators on variable Lebesgue spaces,\nOperator Theory, Pseudo-Differential Equations, and Mathematical Physics,\nOper. Theory Adv. Appl. 228,\nBirkh\u00e4user\/Springer, Basel (2013), 173\u2013183.","DOI":"10.1007\/978-3-0348-0537-7_9"},{"key":"2023033119260594563_j_gmj-2022-2152_ref_012","doi-asserted-by":"crossref","unstructured":"V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko,\nIntegral Operators in Non-Standard Function Spaces. Vol. 1. 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Nekvinda,\nMaximal operator on variable Lebesgue spaces for almost monotone radial exponent,\nJ. Math. Anal. Appl. 337 (2008), no. 2, 1345\u20131365.","DOI":"10.1016\/j.jmaa.2007.04.047"},{"key":"2023033119260594563_j_gmj-2022-2152_ref_016","doi-asserted-by":"crossref","unstructured":"L. Pick and M. R\u016f\u017ei\u010dka,\nAn example of a space \n \n \n \n L\n \n p\n \u2062\n \n (\n x\n )\n \n \n \n \n \n L^{p(x)}\n \n on which the Hardy\u2013Littlewood maximal operator is not bounded,\nExpo. Math. 19 (2001), no. 4, 369\u2013371.","DOI":"10.1016\/S0723-0869(01)80023-2"},{"key":"2023033119260594563_j_gmj-2022-2152_ref_017","doi-asserted-by":"crossref","unstructured":"V. Rabinovich and S. Samko,\nBoundedness and Fredholmness of pseudodifferential operators in variable exponent spaces,\nIntegral Equations Operator Theory 60 (2008), no. 4, 507\u2013537.","DOI":"10.1007\/s00020-008-1566-9"}],"container-title":["Georgian Mathematical Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/gmj-2022-2152\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/gmj-2022-2152\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T06:44:45Z","timestamp":1680331485000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/gmj-2022-2152\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,26]]},"references-count":18,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3,1]]},"published-print":{"date-parts":[[2022,6,1]]}},"alternative-id":["10.1515\/gmj-2022-2152"],"URL":"https:\/\/doi.org\/10.1515\/gmj-2022-2152","relation":{},"ISSN":["1072-947X","1572-9176"],"issn-type":[{"type":"print","value":"1072-947X"},{"type":"electronic","value":"1572-9176"}],"subject":[],"published":{"date-parts":[[2022,3,26]]}}}