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Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension

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 $(-Δ)^{-γ/2}u(x,t)$ 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 $\dot{H}^{-β/2}(\mathbb{R}^n)$ 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 $μ$ on $\mathbb{R}^{n+1}_+$ such that the embeddings $$\|u(\cdot,\cdot)\|_{L^{q_0,p_0}_μ(\mathbb{R}^{n+1})}\lesssim \|f\|_{\dotΛ^{p,q}_β(\mathbb{R}^n)}$$ hold for some $p_0$ and $q_0$ depending on $p$ and $q$ which are classified in three different cases: (1). $p=q\in (n/(n+β),1];$ (2) $(p,q)\in (1,n/β)\times (1,\infty);$ (3). $(p,q)\in (1,n/β)\times\{\infty\}.$ 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.