Function spaces of smoothness zero.

The article focuses on the new spaces of functions of smoothness zero defined on the Euclidean n-space. It states that the spaces suffices to estimate the second term in the embeddings of the spaces' norms. It says that the embedding of the spaces has been proven through the use of integral representations of functions with regards to its differences. It also relates the denoting of Sobolev space and Banach space of functions which was defined with a norm that satisfies a flexible condition. It also discusses several theorems and lemmas implying the integrability of functions with norms.