Symmetric Gaussian Rearrangement with Applications to Gaussian Noise Stability
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We introduce a new symmetric Gaussian rearrangement that applies to sets and functions which are invariant with respect to a given finite subgroup of the orthogonal group. The properties of this rearrangement are developed, connecting it with the existing theory, and we show that the Hardy--Littlewood inequality holds for this rearrangement. As an application, certain integrals involving the Ornstein--Uhlenbeck semigroup are shown to be non-decreasing under rearrangement of their initial data. This in turn demonstrates a new form of Borell's inequality: the (positive correlation) Gaussian noise stability of a set is non-decreasing under appropriate symmetric Gaussian rearrangement.