Supriya Ghosh (Editor)

Hardy–Littlewood inequality

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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

∫ R n f ( x ) g ( x ) d x ≤ ∫ R n f ∗ ( x ) g ∗ ( x ) d x

where f* and g* are the symmetric decreasing rearrangements of f(x) and g(x), respectively.

Proof

From layer cake representation we have:

f ( x ) = ∫ 0 ∞ χ f ( x ) > r d r g ( x ) = ∫ 0 ∞ χ g ( x ) > s d s

where χ f ( x ) > r denotes the indicator function of the subset E f given by

E f = { x ∈ X : f ( x ) > r }

Analogously, χ g ( x ) > s denotes the indicator function of the subset E g given by

E g = { x ∈ X : g ( x ) > s } ∫ R n f ( x ) g ( x ) d x = ∫ R n ∫ 0 ∞ ∫ 0 ∞ χ f ( x ) > r χ g ( x ) > s d r d s d x

References

Hardy–Littlewood inequality Wikipedia
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