1. Rearrangement inequalities on the lattice graph.
- Author
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Gupta, Shubham and Steinerberger, Stefan
- Abstract
The Polya–Szegő inequality in Rn$\mathbb {R}^n$ states that, given a nonnegative function f:Rn→R$f:\mathbb {R}^{n} \rightarrow \mathbb {R}$, its spherically symmetric decreasing rearrangement f∗:Rn→R$f^*:\mathbb {R}^{n} \rightarrow \mathbb {R}$ is ‘smoother’ in the sense of ∥∇f∗∥Lp⩽∥∇f∥Lp$\Vert \nabla f^*\Vert _{L^p} \leqslant \Vert \nabla f\Vert _{L^p}$ for all 1⩽p⩽∞$1 \leqslant p \leqslant \infty$. We study analogues on the lattice grid graph Z2$\mathbb {Z}^2$. The spiral rearrangement is known to satisfy the Polya–Szegő inequality for p=1$p=1$, the Wang‐Wang rearrangement satisfies it for p=∞$p=\infty$ and no rearrangement can satisfy it for p=2$p=2$. We develop a robust approach to show that both these rearrangements satisfy the Polya–Szegő inequality up to a constant for all 1⩽p⩽∞$1 \leqslant p \leqslant \infty$. In particular, the Wang‐Wang rearrangement satisfies ∥∇f∗∥Lp⩽21/p∥∇f∥Lp$\Vert \nabla f^*\Vert _{L^p} \leqslant 2^{1/p} \Vert \nabla f\Vert _{L^p}$ for all 1⩽p⩽∞$1 \leqslant p \leqslant \infty$. We also show the existence of (many) rearrangements on Zd$\mathbb {Z}^d$ such that ∥∇f∗∥Lp⩽cd·∥∇f∥Lp$\Vert \nabla f^*\Vert _{L^p} \leqslant c_d \cdot \Vert \nabla f\Vert _{L^p}$ for all 1⩽p⩽∞$1 \leqslant p \leqslant \infty$. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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