Monotonicity of functionals associated to product measures via their Fourier transform and applications
Andreas Malliaris
TL;DR
This work develops a unified Fourier-analytic framework to study monotonicity properties of functionals of product measures. By relating the monotonicity of $H(a)=\mathbb{E}h(a_1^{1/q}X_1+\cdots+a_n^{1/q}X_n)$ to the log-convexity or log-concavity of $t\mapsto \hat{\phi}(t^{1/q})$, it yields log-convexity when $\hat{\phi}(r^{1/q})$ is log-convex and Schur concavity when it is log-concave; converses are provided. The results unify and extend moment comparisons, Khinchin-type inequalities, and extremal section problems for convex bodies, including high-dimensional bodies like $B_p^n(K)$ and their sections, with applications to intersection bodies and $L_p$ embeddings. The approach yields new sharp inequalities for Gaussian mixtures, $p$-stable/pseudo-stable distributions, and higher co-dimension sections, offering a versatile toolkit for convex-geometry and probabilistic questions. Overall, the paper advances a cohesive Fourier-based method to derive extremal and comparative results across dimensions and norms.
Abstract
Let $μ$ be a probability measure on $\mathbb{R}$. We give conditions on the Fourier transform of its density for functionals of the form $H(a)=\int_{\mathbb{R}^n}h(\langle a,x\rangle)μ^n(dx)$ to be Schur monotone. As applications, we put certain known and new results under the same umbrella, given by a condition on the Fourier transform of the density. These results include certain moment comparisons for independent and identically distributed random vectors, when the norm is given by intersection bodies, and the corresponding vector Khinchin inequalities. We also extend the discussion to higher dimensions.