Transitivity and Unimprovability of Imbedding Theorems. Compactness

Abstract

Suppose given a system of numbers (1) ${\bf\it r} = (r_1,...,r_n) \ > {\bf 0}, \ {\bf\it p} = (p_1,...,p_n) \ (1{\mathop<\limits_=} \ p_l {\mathop<\limits_=}\ \infty)$ and numbers p′, p″, satisfying the inequalities (2) $$p_l \ {\mathop<\limits_=} \ {p^\prime} \ < \ p^{\prime\prime} \ {\mathop<\limits_=}\ \infty.$$ If the following conditions are satisfied : (3) $$\varrho_i^\prime = {r_i\chi\over \chi_i},$$ (4) $$\chi=1-{\mathop\sum\limits_{l=1}^n}{{1\over p_l}-{1\over p^\prime}\over r_l}>0,$$ (5) $$\chi_i=1-{\mathop\sum\limits_{l=1}^n}{{1\over p_l}-{1\over p_i}\over r_l}>0\quad (i=1,...,n),$$ then the imbedding theorem (6.8) holds: $$B_{p\theta}^r({\rm R}_n)\rightarrow B_{p^\prime\theta}^{p^\prime}({\rm R}_n)$$ ccomplishing the transition from the system of numbers (1) to the system of numbers $$\varrho^\prime=(\varrho_1^\prime,...,\varrho_n^\prime), \ p^\prime.$$ 1 S. M. Nikol’skii [3, 10].