The description of functions from anisotropic spaces of complex order and its application (Q1386557)

The authors give some characterization of functions from the anisotropic fractional type function spaces \[ L_{p,r}^{\bar{\alpha}_1,\dots,\bar{\alpha}_n}(R^n)) \bigl\{f: \| f\|_r= +\| F^{-1}m(\xi)Ff\|_p < \infty \bigr\} \] where \( \text{Re } \alpha_j^k \geq 0 \) and \(m(\xi)=\sum_{k=1}^mb_k\prod_{j=1}^n | \xi_j| ^{\alpha_j^k}\), \(b_k\) being arbitrary complex numbers. Here the function \(f\) itself and its anisotropic ``derivative'' \(F^{-1}m(\xi)Ff\) are treated in different \(L_r\) and \(L_p\)-norms. In the case when \(\bar{\alpha}^k=(0,\dots,\alpha_k,\dots,0)\), \(b_k= 1\) and \(r=p\), we have the anisotropic Liouville spaces studied by P. I. Lizorkin. For \(r\neq p\) such spaces were considered in the isotropic case by the reviewer and in the anisotropic case by A. A. Davtyan and G. P. Emgusheva and V. A. Nogin. The general situation treated in the paper covers, in particular, the case of functions with the dominating mixed derivative (P. I. Lizorkin and S. M. Nikolskii). The characterization is given in terms of convergence of the so called approximative operators, which may be considered as some substitution of anisitropic-type hypersingular integrals.