论文标题
Sobolev空间中的大约扩展
Approximate Extension in Sobolev Space
论文作者
论文摘要
令$ l^{m，p}（\ mathbb {r}^n）$是$ p \ in（n，\ infty）$，$μ$的同质sobolev空间，是$ \ m athbb {r}^n $和$ l^n $和$ l^{m，p}的borel常规度量（具有有限seminorm $ \ | f \ | _ {l^{m，p}（\ mathbb {r}^n） + l^p（dμ）}：= \ text {inft} _ {f_1 + f_1 + f_2 = f_2 = f_2 = f_2 = f_2 = f_2 = f_2 = f_2 = f_2 = f_2 \ | f_1 \ | _ {l^{m，p}（\ Mathbb {r}^n）}^p + \ int _ {\ Mathbb {r}^n} | f_2 | f_2 |^pdμ\}^{1/p} $。我们构造一个线性操作员$ t：l^{m，p}（\ mathbb {r}^n） + l^p（dμ）\ to l^{m，p}（\ mathbb {r}^n）$，几乎最佳地将每个功能分解在总和空间中： $ \ | tf \ | _ {l^{m，p}（\ Mathbb {r}^n）}^p + \ int _ {\ Mathbb {\ Mathbb {r}^n} | l^p（dμ）}^p $，$ c $依赖于$ m $，$ n $和$ p $。对于$ e \ subset \ mathbb {r}^n $，让$ l^{m，p}（e）$表示所有限制的空间，in l^{m，p}（\ mathbb {r}^n）$，配备标准痕迹seminorm。对于$ p \ in（n，\ infty）$，我们构造了线性扩展运算符$ t：l^{m，p}（e）\ to l^{m，p}（\ m athbb {r}^n）$满意$ \ | tf \ | _ {l^{m，p}（\ mathbb {r}^n）} \ leq c \ | f \ | _ {l^{m，p}（e）} $，其中$ c $仅取决于$ n $，$ m $和$ p $。我们显示这些运算符可以通过线性功能的集合来表示，其支持的重叠有限。
Let $L^{m,p}(\mathbb{R}^n)$ be the homogeneous Sobolev space for $p \in (n,\infty)$, $μ$ be a Borel regular measure on $\mathbb{R}^n$, and $L^{m,p}(\mathbb{R}^n) + L^p(dμ)$ be the space of Borel measurable functions with finite seminorm $\|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(dμ)} := \text{inf}_{f_1 +f_2 = f} \{ \|f_1\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |f_2|^p dμ\}^{1/p}$. We construct a linear operator $T:L^{m,p}(\mathbb{R}^n) + L^p(dμ) \to L^{m,p}(\mathbb{R}^n)$, that nearly optimally decomposes every function in the sum space: $\|Tf\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |Tf-f|^p dμ\leq C \|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(dμ)}^p$ with $C$ dependent on $m$, $n$, and $p$ only. For $E \subset \mathbb{R}^n$, let $L^{m,p}(E)$ denote the space of all restrictions to $E$ of functions $F \in L^{m,p}(\mathbb{R}^n)$, equipped with the standard trace seminorm. For $p \in (n, \infty)$, we construct a linear extension operator $T:L^{m,p}(E) \to L^{m,p}(\mathbb{R}^n)$ satisfying $Tf|_E = f|_E$ and $\|Tf\|_{L^{m,p}(\mathbb{R}^n)} \leq C \|f\|_{L^{m,p}(E)}$, where $C$ depends only on $n$, $m$, and $p$. We show these operators can be expressed through a collection of linear functionals whose supports have bounded overlap.
