On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

V. P. Il'in

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 20, Tome 201 (1992), pp. 22-94

Résumé

For anisotropic Sobolev and Nikol'skii–Besov spaces with numerical integral-differential (ID) parameters (and for embedding parameters of similar nature), we study the following things: A) the influence of ID parameters on characteristics of conditions determining the possible embeddings; B) the mutual influence of ID parameters and the parameters determining properties of the domain $G$ on the set of admissible values of embedding parameters (AVEP). (By parameters determining properties of the domain $G$ we mean parameters describing the set $\Lambda(G)$ of all vectors $\lambda=(\lambda_1,\dots,\lambda_n)>0$ such that $G$ satisfies the $\lambda$-corn condition; we consider embeddings with $q\geqslant p^i$, where $q$ is the integral embedding parameter and $p^i$'s are the corresponding parameters of the space in question). A). The notion of the initial matrix $A_0$ of the space in question is introduced. This matrix is defined in terms of ID parameters and the smallest (“initial”) values of embedding parameters satisfying the above condition $q\geqslant p^i$. In fact $A_0$ is determined by ID-parameters only, and, what is important, it is a $z$-matrix. A classification of anisotropic spaces is introduced (on the basis of a natural classification for $z$-matrices), that allows one to answer the following questions. Which properties of ID parameters (in other words, of the matrix $A_0$) guarantee the existence of an embedding satisfying $q\geqslant p^i$ (see above)? If such an embedding exist, which form assumes the existence condition, depending on the structure and properties of $A_0$? B). Under the assumption that the initial matrix $A_0$ of the space in question is a non-degenerate $M$-matrix, we investigate mutual influence of $A_0$ (i.e., ID parameters) and $\Lambda(G)$ on the set of AVEP. In this case the existence condition for embedding looks like $A\lambda_G^*\geqslant0$, where $\lambda_G^*\in\Lambda(G)$ is the optimal vector, i.e. the vector such that the inequality just mentioned determines the largest set of AVEP (for given $A_0$ and $\Lambda(G)$). The vector $\lambda_G^*$ can be found by linear optimization methods. The following cases can occur. а) $\lambda_G^*=\lambda_{E^n}^*$ ($\lambda_{E^n}^*$ is the optimal vector associated with the same matrix $A_0$ and the set $\Lambda(E^n)$). b) $\lambda_G^*\ne\lambda_{E^n}^*$. c) $\lambda_G^*$ does not exist. In case a) the set of AVEP is the largest possible, while in case c) there is no embedding satisfying $q\geqslant p^i$. In case b) the set of AVEP is smaller than in case a), and here the “saturation phenomenon” with respect to certain differential parameters occurs (this phenomenon can be described as follows: for some spaces of functions defined on the same domain $G$ that differ by the parameters in question only, the sets of AVEP coincide). This implies some consequences concerning extension of all functions in these spaces from $G$ to $E^n$.