Interpolation Methods for Anisotropic Net Spaces

Abstract

In this paper, we study the interpolation properties of anisotropic net spaces \(N_{\bar{p}, \bar{q}}(M)\), where \(\bar{p} = (p_1, ..., p_n)\), \(\bar{q} = (q_1, ..., q_n)\). It is shown that, with respect to the multidimensional interpolation method, the following equality holds

\[ (N_{\theta}(M), N_{1-\theta}(M))_{\bar{\theta}} = N_{\bar{f}}(M) \quad \text{where} \quad \frac{1}{\bar{p}} = \frac{1 - \bar{\theta}}{\bar{p}_0} + \frac{\bar{\theta}}{\bar{p}_1} \]