论文标题
Poincaré-Sobolev在整个空间上具有重新排列不变规范的不平等现象
Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space
论文作者
论文摘要
提供了整个$ \ Mathbb {r}^n $上重新安排不变规范的Poincaré-Sobolev-type不平等现象。 Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$ are either rearrangement-invariant spaces over $\mathbb{R}^n$或Orlicz的空间$ \ Mathbb {r}^n $，$ u $是$ m $ times弱小的函数，其梯度为$ x $，$ p $最多是$ m-1 $的多项式订单，取决于$ u $，$ u $，$ c $是$ u $的常数独立$ u $。从某种意义上说，当发现空间$ x $时，在这些不平等中的最佳重排空间或Orlicz空间$ y $。还提供了习惯功能空间的各种特定示例。
Poincaré-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$ are either rearrangement-invariant spaces over $\mathbb{R}^n$ or Orlicz spaces over $\mathbb{R}^n$, $u$ is a $m-$times weakly differentiable function whose gradient is in $X$, $P$ is a polynomial of order at most $m-1$, depending on $u$, and $C$ is a constant independent of $u$, are studied. In a sense optimal rearrangement-invariant spaces or Orlicz spaces $Y$ in these inequalities when the space $X$ is fixed are found. A variety of particular examples for customary function spaces are also provided.
