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We determine the exact $L^p(\mathbb{R})$ norm of the centered Hardy--Littlewood maximal operator for every $1<p<\infty$, proving that it equals $p/(p-1)$. This settles the sharp strong-type problem for the centered maximal operator on the real line. For every $p$ in this range, the norm is strictly larger than the cons...
We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p...
We derive sharp $\ell^p(\mathbb{Z})$ bounds for the $k$th derivative of the discrete uncentered maximal operator applied to characteristic functions $f:\mathbb{Z}\to\{0,1\}$ in the cases $k=0,1,2$. When $k=1,2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or disc...
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