A Dyadic Approach to Weak Characterizations of Function Spaces

TL;DR

This paper introduces dyadic-based weak-type quasi-norms for function spaces, establishing new embeddings and characterizations by comparing them with classical spaces, and demonstrating the sharpness of these results through examples.

Contribution

It develops a dyadic approach to weak characterizations of function spaces, providing discrete analogues and new embeddings compared to existing continuous definitions.

Findings

New dyadic weak-type quasi-norms are defined.

Established embeddings between dyadic and classical function spaces.

Examples demonstrate the sharpness of the theoretical results.

Abstract

Weak-type quasi-norms are defined using the mean oscillation or the mean of a function on dyadic cubes, providing discrete analogues and variants of the corresponding quasi-norms on the upper half-space previously considered in the literature. Comparing the resulting function spaces to known function spaces such as $\dot{W}^{1,p}(\rn)$, $\JNp$, $\Lp$ and weak-$\Lp$ gives new embeddings and characterizations of these spaces. Examples are provided to prove the sharpness of the results.