A Dyadic Approach to Weak Characterizations of Function Spaces

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 ${\dot{W}}^{1,p}(\text{rn})$, $\text{JNp}$, $\text{Lp}$ and weak-$\text{Lp}$ gives new embeddings and characterizations of these spaces. Examples are provided to prove the sharpness of the results.