论文标题
晶格图上的重排不平等
Rearrangement Inequalities on the Lattice Graph
论文作者
论文摘要
$ \ mathbb {r}^n $中的polya-szegő不平等说明，给定一个非负函数$ f：\ mathbb {r}^{n}^{n} \ rightarrow \ rightarrow \ mathbb {r} _ {} $，它的球形减小重新安装了重新安排$ f^*：\ math} \ Mathbb {r} _ {} $在$ \ |的意义\ nabla f^*\ | _ {l^p} \ leq \ | \ nabla f \ | _ {l^p} $对于所有$ 1 \ leq p \ leq \ infty $。我们在晶格网格图上研究类似物$ \ mathbb {z}^2 $。已知螺旋重排以满足Polya-szegő不平等，以$ p = 1 $，Wang-Wang重排以$ P = \ infty $满足它，并且没有重新安排可以满足$ P = 2 $的满足。我们开发了一种强大的方法来表明这两个重排都满足了Polya-Szegő不平等，达到了所有$ 1 \ leq p \ leq \ leq \ infty $的常数。特别是，王重排满足$ \ | \ nabla f^*\ | _ {l^p} \ leq 2^{1/p} \ | \ nabla f \ | _ {l^p} $对于所有$ 1 \ leq p \ leq \ infty $。我们还显示了（许多）重新安排在$ \ mathbb {z}^d $上的存在，以便$ \ | \ nabla f^*\ | _ {l^p} \ leq c_d \ cdot \ | \ nabla f \ | _ {l^p} $对于所有$ 1 \ leq p \ leq \ infty $。
The Polya-Szegő inequality in $\mathbb{R}^n$ states that, given a non-negative function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$, its spherically symmetric decreasing rearrangement $f^*:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$ is `smoother' in the sense of $\| \nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We study analogues on the lattice grid graph $\mathbb{Z}^2$. The spiral rearrangement is known to satisfy the Polya-Szegő inequality for $p=1$, the Wang-Wang rearrangement satisfies it for $p=\infty$ and no rearrangement can satisfy it for $p=2$. We develop a robust approach to show that both these rearrangements satisfy the Polya-Szegő inequality up to a constant for all $1 \leq p \leq \infty$. In particular, the Wang-Wang rearrangement satisfies $\| \nabla f^*\|_{L^p} \leq 2^{1/p} \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We also show the existence of (many) rearrangements on $\mathbb{Z}^d$ such that $\| \nabla f^*\|_{L^p} \leq c_d \cdot \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$.
