论文标题
在无限的加权Sobolev函数的限制下
On Limits at Infinity of Weighted Sobolev Functions
论文作者
论文摘要
我们研究了l^1 _ {\ Mathrm {loc}}}（\ Mathbb r^d）$的必要条件$ w \ in l^1 _ {\ mathrm {loc}} $几乎可以确保radial的存在，垂直的限制，无限限制的sobolev函数的sobolev函数$ \ in w^^{1，p} _} r^d，w）$带有$ p $ - 积分梯度$ | \ nabla u | \ in l^p（\ mathbb r^d，w）$。该问题巧妙地取决于限制的意义。 首先，我们完全表征了径向极限的存在。其次，我们为垂直极限的存在提供基本上鲜明的条件。在产品和径向重量的特定设置中，我们仅在语句时才给出。这些概括并为Fefferman和Uspenski \Uı提供新的证据。
We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense in which the limit is taken. First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and radial weights, we give if and only if statements. These generalize and give new proofs for results of Fefferman and Uspenski\uı.
