Hardy–Littlewood inequality
In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean space Rn then
where f* and g* are the symmetric decreasing rearrangements of f and g, respectively.Proof
From layer cake representation we have:
where denotes the indicator function of the subset E f given by
Analogously, denotes the indicator function of the subset E g given by
