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Here, we study unboundedness properties of functions belonging to those spaces in terms of their growth envelopes. This concept has been studied and applied successfully for a variety of smoothness spaces already. Surprisingly, for the generalized Morrey spaces we arrive at three possible cases only, i.e. boundedness, the [Formula: see text]-behavior or the proper Morrey behavior. These cases are characterized in terms of the limit of [Formula: see text] and [Formula: see text] as [Formula: see text] and [Formula: see text], respectively. For the generalized Besov\u2013Morrey spaces the limit of [Formula: see text] as [Formula: see text] also plays a r\u00f4le. <\/jats:p>","DOI":"10.1142\/s0219530523500379","type":"journal-article","created":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T13:59:03Z","timestamp":1703685543000},"page":"751-790","source":"Crossref","is-referenced-by-count":10,"title":["On a bridge connecting Lebesgue and Morrey spaces in view of their growth properties"],"prefix":"10.1142","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2576-0300","authenticated-orcid":false,"given":"Dorothee D.","family":"Haroske","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Friedrich Schiller University Jena, 07737 Jena, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6121-0296","authenticated-orcid":false,"given":"Susana D.","family":"Moura","sequence":"additional","affiliation":[{"name":"University of Coimbra, Department of Mathematics, CMUC, 3000-143 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7484-2900","authenticated-orcid":false,"given":"Leszek","family":"Skrzypczak","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics & Computer Science, Adam Mickiewicz University, ul. 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