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

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 $f$.