# Optimal local embeddings of Besov spaces involving only slowly varying smoothness

## TL;DR

This paper establishes optimal local embeddings for Besov spaces with slowly varying smoothness, extending previous results to new target spaces related to small Lebesgue spaces using advanced interpolation and inequality techniques.

## Contribution

It introduces new optimal embedding results for Besov spaces with slowly varying smoothness, surpassing prior Lorentz-Karamata space targets through innovative methods.

## Key findings

- Derived new local embedding theorems for Besov spaces
- Extended the class of target spaces beyond Lorentz-Karamata spaces
- Applied limiting real interpolation and Hardy inequalities effectively

## Abstract

The aim of the paper is to establish (local) optimal embeddings of Besov spaces $B^{0,b}_{p,r}$ involving only a slowly varying smoothness $b$. In general, our target spaces are outside of the scale of Lorentz-Karamata spaces and are related to small Lebesgue spaces. In particular, we improve results from [CGO11b], where the targets are (local) Lorentz-Karamata spaces. To derive such results, we apply limiting real interpolation techniques and weighted Hardy-type inequalities.