High-dimensional limits and extremizers for maximal functions associated with log-concave densities
We introduce a unified framework to establish the high-dimensional asymptotic behavior of maximal functions associated with radial log-concave probability densities, encompassing the maximal heat semigroup, Hardy-Littlewood maximal function over Euclidean balls, and, additionally, maximal spherical means. Namely, for any $p \in (1, \infty)$, we prove that the $L^p(\mathbb{R}^d)$ operator norms of these maximal operators all converge as the dimension $d \to \infty$ to a single, universal limit $\lambda(p)$. Furthermore, by proving that the $L^p$ operator norms for the heat semigroup $\mathcal G_*^d$ are monotonically non-decreasing in the dimension, we provide explicit quantitative bounds on the universal limit, showing that $\frac{2}{5}\frac{p}{p-1} \le \|\mathcal{G}_*^1\|_{L^p(\mathbb{R}) \to L^p(\mathbb{R})} \le \lambda(p) \le \frac{p}{p-1}$. We also prove an extremality property: among all symmetric convex bodies in high dimensions, the maximal operator associated with the Euclidean ball achieves the asymptotically minimal $L^p$ operator norm. Our main results are established via a general transference principle that allows us to control maximal functions via Fourier multiplier symbols. To estimate these symbols uniformly across log-concave densities, we import variance type bounds and thin-shell type concentration of measure results, which are novel tools in the study of maximal functions. In particular, to prove the extremality property, we require a variance type bound for general log concave measures established in a recent series of breakthroughs in high dimensional convex geometry.