Rearrangement inequalities on the lattice graph

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AbstractThe Polya–Szegő inequality in states that, given a nonnegative function , its spherically symmetric decreasing rearrangement is ‘smoother’ in the sense of for all . We study analogues on the lattice grid graph . The spiral rearrangement is known to satisfy the Polya–Szegő inequality for , the Wang‐Wang rearrangement satisfies it for and no rearrangement can satisfy it for . We develop a robust approach to show that both these rearrangements satisfy the Polya–Szegő inequality up to a constant for all . In particular, the Wang‐Wang rearrangement satisfies for all . We also show the existence of (many) rearrangements on such that for all .