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

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 Hölder functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_α( Mf) \le (1 + α)^{-1}\operatorname{Lip}_α( f)$, $α\in (0,1]$. 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 Hölder functions, the operator norm of the maximal operator is uniformly bounded by $2^{-α/q}$, 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 $p = 1$.