Extending functions from Nikolskii-Besov spaces of mixed smoothness beyond a cube

TL;DR

This paper develops extension operators for Nikolskii and Besov spaces with mixed smoothness, enabling functions defined on a cube to be extended to the whole space while preserving their space properties.

Contribution

It introduces new continuous linear extension operators for Nikolskii and Besov spaces with mixed smoothness from a cube to the entire space.

Findings

Extension operators preserve space properties on the cube.

Coincidence of local and global spaces on the cube.

Framework for extending mixed smoothness spaces.

Abstract

The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity of functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative functions. The author builds continuous linear mappings of such spaces of functions defined on cube $ I^d, $ to ordinary Nikolskii and Besov spaces of mixed smoothness in $ \mathbb R^d, $ that are function extension operators, thus incurring coincidence of both kinds of spaces on the cube $ I^d. $