Hardy–Littlewood maximal function

Summary

Let $f$ be a locally integrable function on $\R^n$. Its Hardy–Littlewood maximal function $Mf$ gives at each point the largest average of $\abs{f}$ over balls centred there. It dominates $\abs{f}$ almost everywhere and controls all local averages of $f$ at once. The maximal operator $f \mapsto Mf$ is bounded on $L^p$ for $1 < p \le \infty$ and of weak type $(1,1)$. These bounds give the Lebesgue differentiation theorem and are a basic tool in harmonic analysis.