{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T04:36:28Z","timestamp":1770698188633,"version":"3.49.0"},"reference-count":54,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,10,9]],"date-time":"2023-10-09T00:00:00Z","timestamp":1696809600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,10,9]],"date-time":"2023-10-09T00:00:00Z","timestamp":1696809600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Anal Math"],"published-print":{"date-parts":[[2023,12]]},"DOI":"10.1007\/s10476-023-0238-y","type":"journal-article","created":{"date-parts":[[2023,10,9]],"date-time":"2023-10-09T09:27:24Z","timestamp":1696843644000},"page":"1007-1039","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nuclear and Compact Embeddings in Function Spaces of Generalised Smoothness"],"prefix":"10.1007","volume":"49","author":[{"given":"D. D.","family":"Haroske","sequence":"first","affiliation":[]},{"given":"H.-G.","family":"Leopold","sequence":"additional","affiliation":[]},{"given":"S. D.","family":"Moura","sequence":"additional","affiliation":[]},{"given":"L.","family":"Skrzypczak","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,9]]},"reference":[{"key":"238_CR1","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1016\/j.jmaa.2004.09.017","volume":"304","author":"A Almeida","year":"2005","unstructured":"A. Almeida, Wavelet bases in generalized Besov spaces, J. Math. Anal. Appl., 304, (2005), 198\u2013211.","journal-title":"J. Math. Anal. Appl."},{"key":"238_CR2","series-title":"Encyclopedia Math. Appl.","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721434","volume-title":"Regular Variation","author":"N H Bingham","year":"1987","unstructured":"N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl., vol. 27, Cambridge University Press (Cambridge, 1987)."},{"key":"238_CR3","unstructured":"M. Bricchi, Tailored function spaces and related h-sets, PhD thesis, Friedrich-Schiller-Universit\u00e4t Jena (2001)."},{"key":"238_CR4","doi-asserted-by":"publisher","first-page":"7","DOI":"10.7169\/facm\/1538186659","volume":"30","author":"M Bricchi","year":"2002","unstructured":"M. Bricchi, Compact embeddings between Besov spaces defined on h-sets, Funct. Approx. Comment. Math., 30, (2002), 7\u201336.","journal-title":"Funct. Approx. Comment. Math."},{"key":"238_CR5","doi-asserted-by":"publisher","first-page":"383","DOI":"10.4171\/ZAA\/1151","volume":"22","author":"M Bricchi","year":"2003","unstructured":"M. Bricchi and S. D. Moura, Complements on growth envelopes of spaces with generalized smoothness in the sub-critical case, Z. Anal. Anwend., 22, (2003), 383\u2013398.","journal-title":"Z. Anal. Anwend."},{"key":"238_CR6","doi-asserted-by":"crossref","first-page":"265","DOI":"10.4171\/zaa\/1289","volume":"25","author":"A M Caetano","year":"2006","unstructured":"A. M. Caetano and W. Farkas, Local growth envelopes of Besov spaces of generalized smoothness, Z. Anal. Anwend., 25, (2006), 265\u2013298.","journal-title":"Z. Anal. Anwend."},{"key":"238_CR7","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1155\/2005\/165785","volume":"3","author":"A M Caetano","year":"2005","unstructured":"A. M. Caetano and D. D. Haroske, Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers, J. Function Spaces Appl., 3, (2005), 33\u201371.","journal-title":"J. Function Spaces Appl."},{"key":"238_CR8","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s00041-006-6018-9","volume":"12","author":"A Caetano","year":"2006","unstructured":"A. Caetano and H. G. Leopold, Local growth envelopes of Triebel\u2013Lizorkin spaces of generalized smoothness, J. Fourier Anal. Appl., 12, (2006), 427\u2013445.","journal-title":"J. Fourier Anal. Appl."},{"key":"238_CR9","doi-asserted-by":"publisher","first-page":"2676","DOI":"10.1016\/j.jfa.2013.03.012","volume":"264","author":"A Caetano","year":"2013","unstructured":"A. Caetano and H. G. Leopold, On generalized Besov and Triebel\u2013Lizorkin spaces of regular distributions, J. Funct. Anal., 264, (2013), 2676\u20132703.","journal-title":"J. Funct. Anal."},{"key":"238_CR10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511897467","volume-title":"Entropy, Compactness and the Approximation of Operators","author":"B Carl","year":"1990","unstructured":"B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge Univ. Press (Cambridge, 1990)."},{"key":"238_CR11","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/j.jat.2017.10.009","volume":"225","author":"F Cobos","year":"2018","unstructured":"F. Cobos, O. Dom\u00ednguez, and Th. K\u00fchn, On nuclearity of embeddings between Besov spaces, J. Approx. Theory, 225, (2018), 209\u2013223.","journal-title":"J. Approx. Theory"},{"key":"238_CR12","doi-asserted-by":"crossref","unstructured":"F. Cobos, D. E. Edmunds, and Th. K\u00fchn, Nuclear embeddings of Besov spaces into Zygmund spaces, J. Fourier Anal. Appl., 26, (2020), Papper No. 9.","DOI":"10.1007\/s00041-019-09709-6"},{"key":"238_CR13","series-title":"Lecture Notes in Math.","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1007\/BFb0078872","volume-title":"Function Spaces and Applications (Lund, 1986)","author":"F Cobos","year":"1998","unstructured":"F. Cobos and D. L. Fernandez, Hardy\u2013Sobolev spaces and Besov spaces with a function parameter, in: Function Spaces and Applications (Lund, 1986), Lecture Notes in Math., vol. 1302, Springer (Berlin, 1998), pp. 158\u2013170."},{"key":"238_CR14","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1016\/j.jat.2007.11.007","volume":"160","author":"F Cobos","year":"2009","unstructured":"F. Cobos and Th. K\u00fchn, Approximation and entropy numbers in Besov spaces of generalized smoothness, J. Approx. Theory, 160, (2009), 56\u201370.","journal-title":"J. Approx. Theory"},{"key":"238_CR15","volume-title":"Spectral Theory and Differential Operators","author":"D E Edmunds","year":"1987","unstructured":"D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press (Oxford, 1987)."},{"key":"238_CR16","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.jat.2016.07.008","volume":"211","author":"D E Edmunds","year":"2016","unstructured":"D. E. Edmunds, P. Gurka, and J. Lang, Nuclearity and non-nuclearity of some Sobolev embeddings on domains, J. Approx. Theory, 211, (2016), 94\u2013103.","journal-title":"J. Approx. Theory"},{"key":"238_CR17","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.jat.2013.11.002","volume":"178","author":"D E Edmunds","year":"2014","unstructured":"D. E. Edmunds and J. Lang, Non-nuclearity of a Sobolev embedding on an interval, J. Approx. Theory, 178, (2014), 22\u201329.","journal-title":"J. Approx. Theory"},{"key":"238_CR18","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1002\/mana.201100195","volume":"286","author":"D E Edmunds","year":"2013","unstructured":"D. E. Edmunds and Y. Netrusov, Entropy numbers of operators acting between vector-valued sequence spaces, Math. Nachr., 286, (2013), 614\u2013630.","journal-title":"Math. Nachr."},{"key":"238_CR19","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511662201","volume-title":"Function Spaces, Entropy Numbers, Differential Operators","author":"D E Edmunds","year":"1996","unstructured":"D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press (Cambridge, 1996)."},{"key":"238_CR20","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/BF02392270","volume":"130","author":"P Enflo","year":"1973","unstructured":"P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math., 130, (1973), 309\u2013317.","journal-title":"Acta Math."},{"key":"238_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10231-004-0110-z","volume":"185","author":"W Farkas","year":"2006","unstructured":"W. Farkas and H.-G. Leopold, Characterisation of function spaces of generalised smoothness, Ann. Mat. Pura Appl., 185, (2006), 1\u201362.","journal-title":"Ann. Mat. Pura Appl."},{"key":"238_CR22","doi-asserted-by":"crossref","unstructured":"A. Grothendieck, Produits tensoriels topologiques et espaces nucl\u00edeaires, Mem. Amer. Math. Soc., 16, (1955), Ch. 1: 196 pp., Ch. 2: 140 pp.","DOI":"10.1090\/memo\/0016"},{"key":"238_CR23","doi-asserted-by":"crossref","unstructured":"D. D. Haroske, H.-G. Leopold, and L. Skrzypczak, Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains, J. Complexity, 69, (2022), Paper No. 01605, 23 pp.","DOI":"10.1016\/j.jco.2021.101605"},{"key":"238_CR24","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.jat.2004.04.008","volume":"128","author":"D D Haroske","year":"2004","unstructured":"D. D. Haroske and S. D. Moura, Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers, J. Approx. Theory, 128, (2004), 151\u2013174.","journal-title":"J. Approx. Theory"},{"key":"238_CR25","doi-asserted-by":"publisher","first-page":"1487","DOI":"10.1016\/j.jfa.2007.12.009","volume":"254","author":"D D Haroske","year":"2008","unstructured":"D. D. Haroske and S. D. Moura, Continuity envelopes and sharp embeddings in spaces of generalized smoothness, J. Funct. Anal., 254 (2008), 1487\u20131521.","journal-title":"J. Funct. Anal."},{"key":"238_CR26","doi-asserted-by":"crossref","unstructured":"D. D. Haroske and L. Skrzypczak, Nuclear embeddings in weighted function spaces, Integral Equations Operator Theory, 92, (2020), Paper No. 46, 37 pp.","DOI":"10.1007\/s00020-020-02603-7"},{"key":"238_CR27","unstructured":"D. D. Haroske and L. Skrzypczak, Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces, arXiv:2211.02594 (2022)."},{"key":"238_CR28","doi-asserted-by":"crossref","unstructured":"D. D. Haroske, L. Skrzypczak, and H. Triebel, Nuclear Fourier transforms, J. Fourier Anal. Appl., 29, (2023), Paper No. 38, 28 pp.","DOI":"10.1007\/s00041-023-10017-3"},{"key":"238_CR29","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1002\/mana.19871330102","volume":"133","author":"G A Kalyabin","year":"1987","unstructured":"G. A. Kalyabin and P. I. Lizorkin, Spaces of generalized smoothness, Math. Nachr., 133, (1987), 7\u201332.","journal-title":"Math. Nachr."},{"key":"238_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-6278-3","volume-title":"Eigenvalue Distribution of Compact Operators","author":"H K\u00f6nig","year":"1986","unstructured":"H. K\u00f6nig, Eigenvalue Distribution of Compact Operators, Birkh\u00e4auser (Basel, 1986)."},{"key":"238_CR31","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1017\/S0013091505000386","volume":"49","author":"T K\u00fchn","year":"2006","unstructured":"Th. K\u00fchn, H.-G. Leopold, W. Sickel, and L. Skrzypczak, Entropy numbers of embeddings of weighted Besov spaces. II, Proc. Edinburgh Math. Soc. (2), 49, (2006), 331\u2013359.","journal-title":"Proc. Edinburgh Math. Soc. (2)"},{"key":"238_CR32","doi-asserted-by":"publisher","first-page":"211","DOI":"10.4171\/ZAA\/1705","volume":"41","author":"T Lamby","year":"2022","unstructured":"T. Lamby and S. Nicolay, Some remarks on the Boyd functions related to the admissible sequences, Z. Anal. Anwend., 41, (2022), 211\u2013227.","journal-title":"Z. Anal. Anwend."},{"key":"238_CR33","series-title":"Lecture Notes in Pure Appl. Math.","first-page":"323","volume-title":"Function Spaces (Poznan\u00ed, 1998)","author":"H-G Leopold","year":"2000","unstructured":"H.-G. Leopold, Embeddings and entropy numbers in Besov spaces of generalized smoothness, in: Function Spaces (Poznan\u00ed, 1998), Lecture Notes in Pure Appl. Math., vol. 213, Marcel Dekker (New York, Basel, 2000), pp. 323\u2013336."},{"key":"238_CR34","unstructured":"H.-G. Leopold, Embeddings for general weighted sequence spaces and entropy numbers, in: V. Mustonen and J. R\u00e1kosn\u00edk, eds., Function Spaces, Differential Operators and Nonlinear Analysis, Proceedings of the Conference held in Sy\u00f6te, June, 1999, Math. Inst. Acad. Sci. Czech Republic, (2000), pp. 170\u2013186."},{"key":"238_CR35","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1515\/GMJ.2000.731","volume":"7","author":"H-G Leopold","year":"2000","unstructured":"H.-G. Leopold, Embeddings and entropy numbers for general weighted sequence spaces: the non-limiting case, Georgian Math. J., 7, (2000), 731\u2013743.","journal-title":"Georgian Math. J."},{"key":"238_CR36","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/j.jmaa.2015.03.072","volume":"429","author":"H-G Leopold","year":"2015","unstructured":"H.-G. Leopold and L. Skrzypczak, Compactness of embeddings of function spaces on quasi-bounded domains and the distribution of eigenvalues of related elliptic operators, part II, J. Math. Anal. Appl., 429, (2015), 439\u2013460.","journal-title":"J. Math. Anal. Appl."},{"key":"238_CR37","first-page":"381","volume-title":"Spaces of generalized smoothness, Appendix to Russian edition of [51]","author":"P I Lizorkin","year":"1986","unstructured":"P. I. Lizorkin, Spaces of generalized smoothness, Appendix to Russian edition of [51], Mir (Moscow, 1986), pp. 381\u2013415 (in Russian)."},{"key":"238_CR38","doi-asserted-by":"crossref","unstructured":"C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in: M. Cwikel and J. Peetre, eds., Interpolation Spaces and Allied Topics in Analysis, Lecture Notes of Math., vol. 1070, Proc. Conf., Lund\/Sweden, 1983, Springer (1984), pp. 183\u2013201.","DOI":"10.1007\/BFb0099101"},{"key":"238_CR39","doi-asserted-by":"crossref","unstructured":"S. D. Moura, Function spaces of generalised smoothness, Dissertationes Math., 398 (2001).","DOI":"10.4064\/dm398-0-1"},{"key":"238_CR40","series-title":"PhD thesis","volume-title":"Function spaces of generalised smoothness, entropy numbers, applications","author":"S D Moura","year":"2001","unstructured":"S. D. Moura, Function spaces of generalised smoothness, entropy numbers, applications, PhD thesis, University of Coimbra, Portugal (2001)."},{"key":"238_CR41","doi-asserted-by":"publisher","first-page":"1190","DOI":"10.1002\/mana.200510545","volume":"280","author":"S D Moura","year":"2007","unstructured":"S. D. Moura, On some characterizations of Besov spaces of generalized smoothness, Math. Nachr., 280, (2007), 1190\u20131199.","journal-title":"Math. Nachr."},{"key":"238_CR42","first-page":"156","volume":"247","author":"O G Parfenov","year":"1997","unstructured":"O. G. Parfenov, Nuclearity of embedding operators from Sobolev classes into weighted spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 247, (1997), 156\u2013165, 301\u2013302 (in Russian); translated in J. Math. Sci. (New York), 101 (2000), 3139\u20133145.","journal-title":"Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)"},{"key":"238_CR43","first-page":"203","volume-title":"Elliptische Differentialgleichungen, Band II, Schriftenreihe Inst. Math. Deutsch. Akad. Wissensch, Berlin, Reihe A, No. 8","author":"A Pietsch","year":"1971","unstructured":"A. Pietsch, r-Nukleare Sobolevsche Einbettungsoperatoren, in: Elliptische Differentialgleichungen, Band II, Schriftenreihe Inst. Math. Deutsch. Akad. Wissensch, Berlin, Reihe A, No. 8, Akademie-Verlag (Berlin, 1971), pp. 203\u2013215."},{"key":"238_CR44","series-title":"North-Holland Mathematical Library","volume-title":"Operator Ideals","author":"A Pietsch","year":"1980","unstructured":"A. Pietsch, Operator Ideals, North-Holland Mathematical Library, vol. 20, North-Holland (Amsterdam, 1980)."},{"key":"238_CR45","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1007\/BF01200378","volume":"7","author":"A Pietsch","year":"1984","unstructured":"A. Pietsch, Grothendieck\u2019s concept of a p-nuclear operator, Integral Equations Operator Theory, 7, (1984), 282\u2013284.","journal-title":"Integral Equations Operator Theory"},{"key":"238_CR46","volume-title":"Eigenvalues and s-numbers","author":"A Pietsch","year":"1987","unstructured":"A. Pietsch, Eigenvalues and s-numbers, Akad. Verlagsgesellschaft Geest & Portig (Leipzig, 1987)."},{"key":"238_CR47","volume-title":"History of Banach Spaces and Linear Operators","author":"A Pietsch","year":"2007","unstructured":"A. Pietsch, History of Banach Spaces and Linear Operators, Birkh\u00e4user Boston Inc. (Boston, MA, 2007)."},{"key":"238_CR48","doi-asserted-by":"publisher","first-page":"95","DOI":"10.4064\/sm-31-1-95-109","volume":"31","author":"A Pietsch","year":"1968","unstructured":"A. Pietsch and H. Triebel, Interpolationstheorie f\u00fcr Banachideale von beschr\u00e4nkten linearen Operatoren, Studia Math., 31 (1968), 95\u2013109.","journal-title":"Studia Math."},{"key":"238_CR49","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1112\/S0024610799007723","volume":"60","author":"V S Rychkov","year":"1999","unstructured":"V. S. Rychkov, On restrictions and extensions of the Besov and Triebel\u2013Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc., 60, (1999), 237\u2013257.","journal-title":"J. London Math. Soc."},{"key":"238_CR50","first-page":"235","volume":"143","author":"A Tong","year":"1969","unstructured":"A. Tong, Diagonal nuclear operators on \u2113p spaces, Trans. Amer. Math. Soc., 143, (1969), 235\u2013247.","journal-title":"Trans. Amer. Math. Soc."},{"key":"238_CR51","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0346-0416-1","volume-title":"Theory of Function Spaces","author":"H Triebel","year":"1983","unstructured":"H. Triebel, Theory of Function Spaces, Birkh\u00fcauser (Basel, 1983); Reprint (Modern Birkh\u00e4user Classics) (2010)."},{"key":"238_CR52","volume-title":"Theory of Function Spaces. III","author":"H Triebel","year":"2006","unstructured":"H. Triebel, Theory of Function Spaces. III, Birkh\u00e4user (Basel, 2006)."},{"key":"238_CR53","doi-asserted-by":"publisher","DOI":"10.4171\/019","volume-title":"Function Spaces and Wavelets on Domains","author":"H Triebel","year":"2008","unstructured":"H. Triebel, Function Spaces and Wavelets on Domains, European Mathematical Society Publishing House (Z\u00fcrich, 2008)."},{"key":"238_CR54","doi-asserted-by":"publisher","first-page":"3038","DOI":"10.1002\/mana.201700049","volume":"290","author":"H Triebel","year":"2017","unstructured":"H. Triebel, Nuclear embeddings in function spaces, Math. Nachr., 290, (2017), 3038\u20133048.","journal-title":"Math. Nachr."}],"container-title":["Analysis Mathematica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10476-023-0238-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10476-023-0238-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10476-023-0238-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,30]],"date-time":"2024-10-30T10:49:50Z","timestamp":1730285390000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10476-023-0238-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,9]]},"references-count":54,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["238"],"URL":"https:\/\/doi.org\/10.1007\/s10476-023-0238-y","relation":{},"ISSN":["0133-3852","1588-273X"],"issn-type":[{"value":"0133-3852","type":"print"},{"value":"1588-273X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,9]]},"assertion":[{"value":"23 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}