On functions of bounded $\Lambda$-variation and integral smoothness

TL;DR

This paper establishes a precise criterion for when integral Lipschitz classes are contained within classes of functions with bounded b1-variation, advancing understanding of function space embeddings.

Contribution

It provides a necessary and sufficient condition for embedding integral Lipschitz classes into b1-variation classes, clarifying the relationship between these function spaces.

Findings

Derived a precise embedding criterion between Lip(; p) and b1 BV classes.

Enhanced understanding of the structure of function spaces with bounded b1-variation.

Connected integral smoothness properties with variation-based function classes.

Abstract

We obtain a necessary and sufficient condition for embeddings of integral Lipschitz classes Lip(\alpha; p) into classes \Lambda BV of functions of bounded \Lambda-variation.