On the Embedding of $BV$ Space into Besov-Orlicz Space

TL;DR

This paper establishes a condition for when functions with integrable gradients in BV space can be embedded into Besov-Orlicz spaces, using integral inequalities and molecular decomposition techniques.

Contribution

It provides a new sufficient and necessary condition for embedding BV into Besov-Orlicz spaces, involving simple integral inequalities and molecular decomposition methods.

Findings

Derived a necessary and sufficient embedding condition for BV into Besov-Orlicz spaces.

Introduced an example with Matuszewska-Orlicz indices equal to one.

Utilized molecular decomposition of BV functions as a key tool.

Abstract

We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.