On the Trace of $\dot{W}_{a}^{m+1,1}(\mathbb{R}_{+}^{n+1})$

TL;DR

This paper establishes new extension results for functions in Besov spaces, especially for the case p=1, providing integral estimates for heat extensions in both homogeneous and inhomogeneous settings.

Contribution

It introduces novel extension theorems for Besov spaces in the homogeneous setting and offers new proofs for classical results applicable to inhomogeneous spaces.

Findings

Proves integral estimates for heat extensions of Besov space functions.

Extends classical results to the case p=1 with new proof techniques.

Provides homogeneous and inhomogeneous setting results.

Abstract

In this paper we prove extension results for functions in Besov spaces. Our results are new in the homogeneous setting, while our technique applies equally in the inhomogeneous setting to obtain new proofs of classical results. While our results include $p>1$, of principle interest is the case $p=1$, where we show that \begin{equation*} \int_{\mathbb{R}_{+}^{n+1}}t^{a}|\nabla^{m+1}u(x,t)|\;dtdx\lesssim\left\vert f\right\vert _{B^{m-a,1}(\mathbb{R}^{n})} \end{equation*} for all $f \in \dot{B}^{m-a,1}(\mathbb{R}^{n})$ (the homogeneous Besov space) where $u$ is a suitably scaled heat extension of $f$.