论文标题
通过Caffarelli-Silvestre扩展，能力和Wolff电位的功能空间的嵌入
Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials
论文作者
论文摘要
令$p_αf（x，t）$为平滑函数$ f（x）的caffarelli-silvestre扩展：\ mathbb {r}^n \ rightArrow \ rightArrow \ Mathbb {r}^{n+1} _+：_+：= \ mathbb {r}^n \ times（r}^n \ times（0，\ iffty）。首先，我们想表征$ \ mathbb {r}^{n+1} _+$上的非负量值$μ$，以便$ f（x）\ rightArrowp_αf（x，x，x，t）$诱导了lebesgue space space spaces $ l^p（ $ l^q（\ mathbb {r}^{n+1} _+，μ）。$一方面，这些嵌入将以使用与Caffarelli-Silvestre Extension相关的新引入的$ l^p- $容量来表征。在此过程中，将分别确定$p_αF（x，t）的混合规范估计，$ l^p- $容量的双重形式，一般球的$ l^p- $容量和电容性强类型不平等。另一方面，当$ p> q> 1时，$这些嵌入也将以$μ的hedberg-wolff潜力来表征。其次，我们表征了$ \ mathbb {r}^{r}^{n+1} _} y y $ fr（x）\（x）\ rightarrow p _ p _ f（x）的非负量度$μ$ $ \ mathbb {r}^{r}^{n+1}同质sobolev space $ \ dot {w}^{β，p}（\ m athbb {r}^n）$ to $ l^q（\ m athbb {r}^{r}^{n+1} _+，μ）$在开放式案例和终点案例的分数方面，对于终点案例和一般情况。
Let $P_α f(x,t)$ be the Caffarelli-Silvestre extension of a smooth function $f(x): \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}_+:=\mathbb{R}^n\times (0,\infty).$ The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure $μ$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_α f(x,t)$ induces bounded embeddings from the Lebesgue spaces $L^p(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,μ).$ On one hand, these embeddings will be characterized by using a newly introduced $L^p-$capacity associated with the Caffarelli-Silvestre extension. In doing so, the mixed norm estimates of $P_α f(x,t),$ the dual form of the $L^p-$capacity, the $L^p-$capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when $p>q>1,$ these embeddings will also be characterized in terms of the Hedberg-Wolff potential of $μ.$ Secondly, we characterize a nonnegative measure $μ$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_α f(x,t)$ induces bounded embeddings from the homogeneous Sobolev spaces $\dot{W}^{β,p}(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,μ)$ in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.
