View article

72 A. NAJAFOV for j∈ e; ke j= 0 for j∈ en\e= e′;[tj] 1= min {1, tj}, for j∈ en; and let h0, t0∈(0,∞) n be fixed positive vectors and be∫ ae f (x) dxe=(∏ j∈ e bj∫ aj dxj) f (x), ie, the integration takes place only with respect to the variable xj, whose indices belong to the set e. We say that the open set G⊂ Rn satisfies the condition (A1), if for any x∈ G and T∈(0,∞) n there exists the vector-function ρ (t, x)=(ρ1 (t1, x), ρ2 (t2, x),..., ρn (tn, x)), 0≤ tj≤ Tj, j∈ en, with the following properties:

1) for all j∈ en, the functions ρj (tj, x) are absolutely continuous with respect to tj on [0, Tj], and∣

∣ ρ′ j (tj, x)∣∣≤ 1 for almost all tj∈[0, Tj], where ρ′ j (tj, x)=∂∂ tj ρj (tj, x); 2) ρj (0, x)= 0 for all j∈ en, x+ V (x, ω)= x+⋃

0≤ tj≤ Tj, j∈ en [ρ (t, x)+ tωI]⊂ G, where ω=(ω1, ω2,..., ωn), ωj∈(0, 1] for j∈ en, I=[− 1, 1] n, tωI={(t1ω1y1, t2ω2y2,..., tnωnyn): y∈ I}. If t1= tλ1,..., tn= tλn, λ=(λ1, λ2,..., λn), ρ (t, x)= ρ (tλ, x), ωλ=(ωλ1, ωλ2,..., ωλn), ω∈(0, 1], then V (λ, x, ω)=⋃

0≤ t≤ T [ρ (tλ, x)+ tλωλI] is a flexible λ-horn introduced by OV Besov [11].

Definition. By the space of Becov-Morrey type with the dominant mixed derivatives Sl p, θ, a, æ, τ B (Gh) is meant the Banach space of locally summable on G functions f with the finite norm (mj> lj− kj> 0, j∈ en): fSl p, θ, a, æ, τ B (Gh)=∑ e⊆ en {he 0∫ 0e [∥∥∆ me (h, Gh) Dke f∥∥ p, a, æ, τ