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