The embedding theorem for Sobolev spaces in Lebesgue spaces was proved by S. L. Sobolev in 1938 for domains with the cone condition, and then generalized to the case of domains and norms of spaces like Sobolev spaces of a more general form. The report will consider anisotropic (in terms of the order of derivatives and summability exponents) spaces of the Sobolev type on domains with the flexible σ-cone condition (such domains were first considered in 2001 by O. V. Besov).
Necessary and sufficient conditions for the fulfillment of the corresponding additive and multiplicative estimates (embeddings and multiplicative inequalities of the type of Gagliardo-Nirenberg inequalities) will be considered. It is also planned to consider the question of the density of the set of smooth functions in weighted anisotropic spaces like Sobolev spaces.
