Description of the rearrangement-invariant hull of an anisotropic Calderón space

Introduction (translated from the Russian): "We consider an anisotropic Calderón-type space Lambda (E,overrightarrow F), overrightarrow F={F_1,dots,F_n}. In a rearrangement-invariant space (RIS) E=E({bf R}^n) it is defined as a subspace that consists of functions fin E such that their best approximations e_t^{(j)}(f)_E in the norm of E by entire functions of exponential type of degree t>0 in the jth variable belong (as functions of t) to a Banach function space (BFS) F_j=F_j({bf R}_+), {bf R}_+=(0,infty). This concept includes the classical anisotropic spaces B_{pq}^{overrightarrow alpha}({bf R}^n) introduced by S. M. Nikolʹskiĭ and O. V. Besov and various spaces of generalized smoothness. Similar spaces were introduced by Calderón, and their theory was developed further by K. K. Golovkin, Yu. A. Brudnyĭ and V. K. Shalashov, and Golʹdman. For isotropic Calderón spaces, the problem of optimal embedding into an RIS was solved by Golʹdman and R. Kerman [Tr. Mat. Inst. Steklova {bf 243} (2003), Funkts. Prostran., Priblizh., Differ. Uravn., 161--193; [msn] MR2049469 (2005b:46070) [/msn]]. Golʹdman and Ènrikes ["Optimal cone for rearrangements of functions from an anisotropic Calderón space" (Russian), Moscow, 2003 (manuscript deposited at VINITI, Deposition No. 2221-V); per bibl.] described the cone of decreasing rearrangements for functions from an anisotropic Calderón space and established a criterion for embedding into an RIS X=X({bf R}^n): Lambda(E,overrightarrow F)subset Xtag1 (without a priori constraints on X). In the present paper, we construct an optimal (the tightest) RIS X_0 for the embedding (1), i.e., an RIS such that the embedding (1) is valid for X=X_0 and if X is an RIS satisfying (1), then X_0subset X. Yu. V. Netrusov [Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) {bf 159} (1987), Chisl. Metody i Voprosy Organiz. Vychisl. 8, 69--82, 177; [msn] MR0885077 (88e:46030) [/msn]; Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) {bf 159} (1987), Chisl. Metody i Voprosy Organiz. Vychisl. 8, 103--112, 178; [msn] MR0885079 (88h:46073) [/msn]] obtained a similar formulation of the problem of optimal embedding for some generalized Besov and Lizorkin-Triebel spaces. In the present paper we apply the results obtained to generalized anisotropic Besov spaces. par "The paper has five sections. In §1, we introduce the main notation and define the Caldéron-type spaces investigated. In §2, we give results on absorption and embedding of cones of nonnegative functions into a BFS, as well as the construction of a minimal RIS that contains a given cone of nonnegative decreasing functions. In §3, based on these results, we construct a minimal RIS for the embedding (1). In §4, we obtain an equivalent description for an optimal RIS. §5 is devoted to the application of these results to anisotropic generalized Besov spaces.