The Hardy -- Littlewood maximal operator appears in many places .
Some of its most notable uses are in the proofs of the Lebesgue differentiation theorem and Fatou 's theorem and in the theory of singular integral operators .
The Hardy -- Littlewood maximal operator appears in many places but some of its most notable uses are in the proofs of the Lebesgue differentiation theorem and Fatou 's theorem and in the theory of singular integral operators .