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 $$(-\varDelta )^{-\gamma /2}u(x,t)$$ 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 $$\begin{aligned} \Vert u(\cdot ,\cdot )\Vert _{L^{q_0,p_0}_{\mu }({\mathbb {R}}^{n+1})}\lesssim \Vert f\Vert _{{\dot{\varLambda }}^{p,q}_\beta ({\mathbb {R}}^n)} \end{aligned}$$ 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 fractional Besov capacity of open sets.