Hardy-Littlewood maximal operator

The Hardy-Littlewood maximal operator in n is an operator defined on Lloc1(n) (the space of locally integrable functions in n with the Lebesgue measureMathworldPlanetmath) which maps each locally integrable function f to another function Mf, defined for each xn by


where the supremum is taken over all cubes Q containing x. This function is lower semicontinuous (and hence measurable), and it is called the Hardy-Littlewood maximal function of f.

The operator M is sublinear, which means that


for each pair of locally integrable functions f,g and scalars a,b.

Title Hardy-Littlewood maximal operator
Canonical name HardyLittlewoodMaximalOperator
Date of creation 2013-03-22 13:27:30
Last modified on 2013-03-22 13:27:30
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 8
Author azdbacks4234 (14155)
Entry type Definition
Classification msc 28A25
Classification msc 28A15
Related topic HardyLittlewoodMaximalTheorem
Defines Hardy-Littlewood maximal function