A. H. Kalidolday | Searcharxiv

arXiv2021

Interpolation properties of certain classes of net spaces

The paper studies the interpolation properties of net spaces $N_{p,q}(M)$, when $M$ is the set of dyadic cubes in $\mathbb{R}^n$, and also when $M$ is the family of all cubes with parallel faces to the coordinate axes in $\mathbb{R}^n$. It is shown that, in the case when $M$ is the set of dyadic cubes the scale of spaces is closed with respect to the real interpolation method. In the case, when $M$ is the set of all cubes with parallel faces to the coordinate axes, an analogue of the Marcinkiewicz-Calderon theorem on cones of non-negative functions is given.

math.FA↗

arXiv2020

Interpolation theorem for anisotropic net spaces

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 $$ (N_{\bar{p}_0,\bar{q}_0}(M), N_{\bar{p}_1,\bar{q}_1}(M))_{\barθ,\bar{q}}=N_{\bar{p},\bar{q}}(M),\;\;\; \frac{1}{\bar{p}}=\frac{1-\barθ}{\bar{p}_0}+\frac{\barθ}{\bar{p}_1}. $$

math.CA↗