  Let $u(\cdot,\cdot)$ be the Caffarelli-Silvestre extension of $f.$ The first
goal of this article is to establish the fractional trace type inequalities
involving the Caffarelli-Silvestre extension $u(\cdot,\cdot)$ of $f.$ In doing
so, firstly, we establish the fractional Sobolev/ logarithmic Sobolev/ Hardy
trace inequalities in terms of $\nabla_{(x,t)}u(x,t).$ Then, we prove the
fractional anisotropic Sobolev/ logarithmic Sobolev/ Hardy trace inequalities
in terms of $ {\partial_{t} u(x,t)}$ or $(-\Delta)^{-\gamma/2}u(x,t)$ only.
Moreover, based on an estimate of the Fourier transform of the
Caffarelli-Silvestre extension kernel and the sharp affine weighted $L^p$
Sobolev inequality, we prove that the $\dot{H}^{-\beta/2}(\mathbb{R}^n)$ norm
of $f(x)$ can be controlled by the product of the weighted $L^p-$affine energy
and the weighted $L^p-$norm of ${\partial_{t} u(x,t)}.$ The second goal of this
article is to characterize non-negative measures $\mu$ on $\mathbb{R}^{n+1}_+$
such that the embeddings
$$\|u(\cdot,\cdot)\|_{L^{q_0,p_0}_{\mu}(\mathbb{R}^{n+1})}\lesssim
\|f\|_{\dot{\Lambda}^{p,q}_\beta(\mathbb{R}^n)}$$ hold for some $p_0$ and $q_0$
depending on $p$ and $q$ which are classified in three different cases: (1).
$p=q\in (n/(n+\beta),1];$ (2) $(p,q)\in (1,n/\beta)\times (1,\infty);$ (3).
$(p,q)\in (1,n/\beta)\times\{\infty\}.$ For case (1), the embeddings can be
characterized in terms of an analytic condition of the variational capacity
minimizing function, the iso-capacitary inequality of open balls, and other
weak type inequalities. For cases (2) and (3), the embeddings are characterized
by the iso-capacitary inequality for fractonal Besov capacity of open sets.
