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

Abstract

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 $$\Vert u(\cdot ,\cdot ){\Vert }_{L_{\mu }^{q_0,p_0}({\mathbb{R}}^{n+1})}\lesssim \Vert f{\Vert }_{{\dot{\Lambda }}_{\beta }^{p,q}({\mathbb{R}}^n)}$$ 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.

The case of $p\ge 1$ is added