On Equivalent Norms of Anisotropic Hоlder-Zygmund Spaces of Variable Smoothness

Anisotropic Holder-Zygmund spaces (in the general case, a distribution) defined on Rn with a variable smoothness index, which can also take negative values, are introduced and studied. To define these spaces, a partition of the Littlewood-Paley unit modified for the anisotropic case was used, with the help of which a norm was introduced in an anisotropic space by analogy with an isotropic one. In the case when the closure of the domain of values of the smoothness index contains only positive values, a family of norms in an anisotropic Holder-Zygmund space with a variable smoothness index was constructed using various powers of the finite difference operator. The equivalence of the family of norms defined using finite differences with the norm defined using the partition of the Littlewood-Paley unit on the introduced anisotropic spaces is proved.