Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function

TL;DR

This paper derives sharp bounds for the modulus of continuity of the uncentered Hardy-Littlewood maximal function, providing explicit constants for Lipschitz and Hölder functions, and analyzes high-dimensional asymptotic behavior for various geometric shapes.

Contribution

It introduces integral formulas to obtain optimal bounds on the maximal function's modulus of continuity, including explicit constants and high-dimensional asymptotics for different shapes.

Findings

Best Lipschitz constant on subintervals: (1 + α)^{-1}

Global Lipschitz bound: (√2 - 1)

High-dimensional bounds approach 2^{-α/q} as dimension increases

Abstract

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and H\"older functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_\alpha ( Mf) \le (1 + \alpha)^{-1}\operatorname{Lip}_\alpha( f)$, $\alpha\in (0,1]$. On $\mathbb{R}$, the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le (\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the asymptotic behavior, as $d\to\infty$, of the norm of the maximal operator associated to cross-polytopes, euclidean balls and cubes, that is, $\ell_p$ balls for $p = 1, 2, \infty$. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and H\"older functions, the operator norm of the maximal operator is uniformly bounded by $2^{-\alpha/q}$, where $q$ is the conjugate exponent of $p=1,2$, and as $d\to\infty$ the norms approach this bound. When $p=\infty$, best constants are the same as when $p = 1$.