Resolution of open problems via Orlicz-Zygmund spaces and new geometric properties of Morrey spaces in the Besov sense with non-standard growth

In this paper, we introduced a novel norm structure and corresponding modular for Orlicz-Zygmund spaces, designed to capture summability and integrability properties under non-standard growth conditions. Moreover, we established the Hermite-Hadamard inequalities and gave a positive answer to the open problem 3.11 of Almeida and Hästö in their article (Besov spaces with variable smoothness and integrability, J. Funct. Anal., 258 (2010), 1628–1655), since the Hermite-Hadamard inequalities improve triangular-type inequalities. In addition, we explored some new structural properties of Besov-type Morrey spaces of summability and integrability. Furthermore, we addressed an open problem concerning the compactness of Morrey-type spaces characterized by summability and integrability properties. This problem was originally posed by Peter Hästö during the conference 'Nonstandard Growth Phenomena', held in Turku, Finland, from August 29 to 31, 2017.