Interpolation properties of certain classes of net spaces

TL;DR

This paper investigates the interpolation properties of net spaces defined on dyadic and axis-aligned cubes in bR^n, revealing their closure under real interpolation and extending classical theorems to cone settings.

Contribution

It establishes the interpolation behavior of net spaces on different cube families and extends the Marcinkiewicz-Calderon theorem to cones of non-negative functions.

Findings

Net spaces with dyadic cubes are closed under real interpolation.

An analogue of the Marcinkiewicz-Calderon theorem is proved for cones of non-negative functions.

The results extend classical interpolation theorems to new geometric settings.

Abstract

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.