Sobolev embeddings in grand variable Herz-Morrey Besov spaces

This paper develops a comprehensive framework for the study of grand variable Herz-Morrey Besov spaces with variable smoothness and integrability. We begin by introducing the fundamental notation and preliminary concepts required for the analysis. Building on these foundations, we establish several key analytical tools, including a generalized Plancherel-Polya-Nikolskij inequality adapted to the structure of the grand variable Herz-Morrey spaces, extending classical results previously formulated in the setting of variable Lebesgue spaces. To maintain clarity of exposition, the technical proofs of these auxiliary results are presented separately. We then define the grand variable Herz-Morrey Besov spaces and investigate their structural properties. The final part of the paper is devoted to embedding results, where we establish a Sobolev-type embedding theorem within this generalized framework and discuss its analytical consequences. These findings contribute to the further development of harmonic analysis in variable exponent and Herz-type function space settings.