Interpolation theorem for anisotropic net spaces

TL;DR

This paper establishes an interpolation theorem for anisotropic net spaces, demonstrating how these spaces behave under multidimensional interpolation methods, which advances the understanding of their structural properties.

Contribution

The paper proves a new interpolation equality for anisotropic net spaces, extending the theoretical framework of these function spaces.

Findings

Interpolation equality for anisotropic net spaces established

Multidimensional interpolation method applied successfully

Results enhance understanding of anisotropic space structures

Abstract

The paper studies the interpolation properties of anisotropic net spaces $N_{\bar{p},\bar{q}}(M)$, where $\bar{p}=(p_1, p_2)$, $\bar{q}=(q_1, q_2)$. It is shown that the following equality holds with respect to the multidimensional interpolation method $$ (N_{\bar{p}_0,\bar{q}_0}(M), N_{\bar{p}_1,\bar{q}_1}(M))_{\bar{\theta},\bar{q}}=N_{\bar{p},\bar{q}}(M),\;\;\; \frac{1}{\bar{p}}=\frac{1-\bar{\theta}}{\bar{p}_0}+\frac{\bar{\theta}}{\bar{p}_1}. $$