论文标题
Sobolev规范的令人惊讶的公式
A surprising formula for Sobolev norms
论文作者
论文摘要
我们建立了Sobolev Semi-Norm $ \ | \ nabla u \ | _ {l^p} $与在将强$ l^p $替换为弱$ l^p $中获得的数量时，在gagliardo semi-norm中的弱$ l^p $ norm中获得的数量。作为推论，我们在某些特殊情况（涉及$ w^{1,1} $）中得出了替代性估计，其中“预期”部分Sobolev和Gagliardo-Nirenberg不平等失败了。
We establish the equivalence between the Sobolev semi-norm $\|\nabla u\|_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak $L^p$ norm in the Gagliardo semi-norm $|u|_{W^{s,p}}$ computed at $s = 1$. As corollaries we derive alternative estimates in some exceptional cases (involving $W^{1,1}$) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail.
