# Traces of Besov spaces revisited

## TL;DR

This paper revisits the trace properties of Besov spaces on hyperplanes, especially in borderline cases with $s$ near critical values, revealing new dependencies on the sum-exponent $q$ and simplifying classical results.

## Contribution

It provides new analysis of Besov space traces at critical smoothness levels, introduces a simple mixed-norm estimate approach, and simplifies proofs of classical borderline surjectivity results.

## Key findings

- New dependence on sum-exponent $q$ in borderline Besov trace cases
- Restriction operator for $s$ down to $1/p$ is unsuitable for elliptic problems
- Simplified proof of surjectivity at $s=1/p$ for $q\in(0,1]$

## Abstract

For the trace of Besov spaces $B^s_{p,q}$ onto a hyperplane, the borderline case with $s=\frac{n}{p}-(n-1)$ and $0<p<1$ is analysed and a new dependence on the sum-exponent $q$ is found. Through examples the restriction operator defined for $s$ down to $1/p$, and valued in $L_p$, is shown to be distinctly different and, moreover, unsuitable for elliptic boundary problems. All boundedness properties (both new and previously known) are found to be easy consequences of a simple mixed-norm estimate, which also yields continuity with respect to the normal coordinate. The surjectivity for the classical borderline $s=\frac1p$ ($1\le p<\infty$) is given a simpler proof for all $q\in\,]0,1]$, using only basic functional analysis. The new borderline results are based on corresponding convergence criteria for series with spectral conditions.