Research Problems in Function Theory — Problem 7.74

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\ge1$ and let $B\subset\mathbb R^n$ be a compact convex body with nonempty interior, $B=-B$ and volume $1$. Define centered maximal operators \[M_Bf(x)=\sup_{t>0}t^{-n}\int_{tB}|f(x+y)|\,dy, \qquad \widetilde M_Bf(x)=\sup_{k\in\mathbb Z}2^{-kn}\int_{2^kB}|f(x+y)|\,dy. \] The weak $(1,1)$ constant is the least $C$ for which $|\{x:M_Bf(x)>\lambda\}|\le C\lambda^{-1}\|f\|_1$ for every integrable $f$ and $\lambda>0$; define its lacunary counterpart in the same way. The strong $(p,p)$ constant is the operator norm on $L^p(\mathbb R^n)$.

Statement recorded in the supplied source.

Answer all four questions: (a) for volume-one Euclidean balls, is the weak $(1,1)$ constant of $\widetilde M_B$ bounded independently of $n$? (b) Is the corresponding assertion true for $M_B$? (c) In dimension $1$, with $B=[-1/2,1/2]$, is the sharp weak $(1,1)$ constant of $M_B$ equal to $3/2$? (d) For each fixed $1<p\le3/2$, can $\|M_Bf\|_p\le C_p\|f\|_p$ hold with $C_p$ independent of both $n$ and $B$; does it at least hold uniformly in $n$ for $B=[-1/2,1/2]^n$? The source update records counterexamples to the conjectured weak-type constant and investigations of its sharp value; part (c) is a historical conjecture, not a claimed theorem. [S2]

Short English statement

Determine dimension-free weak bounds for ball averages, the historical one-dimensional sharp constant, and dimension-and-body-free strong bounds in the stated exponent range.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307074 (AMR-022-7074). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.74, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.