About unimprovability the embedding theorems for anisotropic Nikol'skii-Besov spaces with dominated mixed derivates and mixed metric and anisotropic Lorentz spaces

The embedding theory of spaces of differentiable functions of many variables studies important connections and relationships between differential (smoothness) and metric properties of functions and has wide application in various branches of pure mathematics and its applications. Earlier, we obtained the embedding theorems of different metrics for Nikol'skii-Besov spaces with a dominant mixed smoothness and mixed metric, and anisotropic Lorentz spaces. In this work, we showed that the conditions for the parameters of spaces in the above theorems are unimprovable. To do this, we built the extreme functions included in the spaces from the left sides of the embeddings and not included in the "slightly narrowed" spaces from the spaces in the right parts of the embeddings.