The periodic Gagliardo seminorm is nonincreasing under periodic and cylindrical rearrangements for all 1 ≤ p < ∞, and equality forces the function to coincide with its rearrangement up to translation and sign.
Brock, A general rearrangement inequality ` a la Hardy-Littlewood , J
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Strict rearrangement inequalities: nonexpansivity and periodic Gagliardo seminorms
The periodic Gagliardo seminorm is nonincreasing under periodic and cylindrical rearrangements for all 1 ≤ p < ∞, and equality forces the function to coincide with its rearrangement up to translation and sign.