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arXiv 2011.10855math.FAmath.CA
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索伯列夫空间的近似扩张

Approximate Extension in Sobolev Space

Marjorie K. Drake

备注：
arXiv admin note: text overlap with arXiv:1205.2525 by other authors
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摘要：

设$L^{m,p}(\mathbb{R}^n)$是$p \in (n,\infty)$的齐次索伯列夫空间，$μ$是$\mathbb{R}^n$上的Borel正则测度，$L^{m,p}(\mathbb{R}^n) + L^p(dμ)$是具有有限个Borel可测函数的空间$\|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(dμ)} := \text{inf}_{f_1 +f_2 = f} \{ \|f_1\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |f_2|^p dμ\}^{1/p}$ .我们构造了一个线性算子$T:L^{m,p}(\mathbb{R}^n) + L^p(dμ) \to L^{m,p}(\mathbb{R}^n)$，它几乎最优地分解和空间中的每个函数：$\|Tf\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |Tf-f|^p dμ\leq C \|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(dμ)}^p$，其中$C$仅依赖于$m$，$n$和$p$。对于$E \subset \mathbb{R}^n$，令$L^{m,p}(E)$表示对函数$F \in L^{m,p}(\mathbb{R}^n)$的$E$的所有限制的空间，其配备有标准跟踪器。对于$p \in (n, \infty)$，我们构造了一个满足$Tf|_E = f|_E$和$\|Tf\|_{L^{m,p}(\mathbb{R}^n)} \leq C \|f\|_{L^{m,p}(E)}$的线性扩张算子$T:L^{m,p}(E) \to L^{m,p}(\mathbb{R}^n)$，其中$C$只依赖于$n$，$m$和$p$ .我们表明，这些运营商可以表示通过一系列的线性泛函的支持有界重叠。

英文摘要：

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.

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