论文标题
通过Caffarelli-Silvestre扩展
Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension
论文作者
论文摘要
令$ u（\ cdot，\ cdot）$为$ f。$的caffarelli-silvestre扩展本文的第一个目标是建立涉及caffarelli-silvestre扩展$ u（\ cdot，\ cdot，\ cdot，\ cdot）$ f。$ f。hend of first sheinly sobore sobore soboles soboles的分数痕量类型的不平等现象，不等式的不等式_ {（x，t）} u（x，x，t）。 仅有的。此外，基于对咖啡雷利 - 丝绒扩展的傅里叶变换的估计，以及急剧加权的$ l^p $ sobolev不平等，我们证明$ \ dot {h}^h}^{ - β/2}（\ mathb {r}^n）$ f（r}^n）$ f（r}^n）$ f（x）$ f（x）$ f（x）$ f（x）$ f（x）加权$ l^p- $ quarm of $ {\ partial_ {t} u（x，x，t）}。$本文的第二个目标是表征非负测度$μ$ $ \ mathbb {r}^{n+1} _++ $ \ | u（\ cdot，\ cdot）\ | _ {l^{q_0，p_0}_μ（\ Mathbb {r}^{n+1}）} \ lyseSim \ simsim \ | f \ | _ _ {\ dotome $ p_0 $和$ q_0 $，具体取决于$ p $和$ q $，在三种不同的情况下分类：（1）。 $ p = q \ in（n/（n+β），1]； $（2）$（p，q）差异能力最小化功能，开放球的异常不平等以及其他弱类型的不平等。
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.
