Hardy–Littlewood inequality

In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f {\displaystyle f} and g {\displaystyle g} are nonnegative measurable real functions vanishing at infinity that are defined on n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , then ∫ R n f ( x ) g ( x ) d x ≤ ∫ R n f ∗ ( x ) g ∗ ( x ) d x {\displaystyle \int _{\mathbb {R} ^{n}}f(x)g(x)\,dx\leq \int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx} where f ∗ {\displaystyle f^{*}} and g ∗ {\displaystyle g^{*}} are the symmetric decreasing rearrangements of f {\displaystyle f} and g {\displaystyle g} , respectively. The decreasing rearrangement f ∗ {\displaystyle f^{*}} of f {\displaystyle f} is defined via the property that for all r > 0 {\displaystyle r>0} the two super-level sets E f ( r ) = { x ∈ X : f ( x ) > r } {\displaystyle E_{f}(r)=\left\{x\in X:f(x)>r\right\}\quad } and E f ∗ ( r ) = { x ∈ X : f ∗ ( x ) > r } {\displaystyle \quad E_{f^{*}}(r)=\left\{x\in X:f^{*}(x)>r\right\}} have the same volume ( n {\displaystyle n} -dimensional Lebesgue measure) and E f ∗ ( r ) {\displaystyle E_{f^{*}}(r)} is a ball in R n {\displaystyle \mathbb {R} ^{n}} centered at x = 0 {\displaystyle x=0} , i.e.

Source: Wikipedia — Hardy–Littlewood inequality (CC BY-SA 4.0)

Hardy–Littlewood inequality

In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f {\displaystyle f} and g {\displaystyle g} are nonnegative measurable real functions vanishing at infinity that are defined on n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , then ∫ R n f ( x ) g ( x ) d x ≤ ∫ R n f ∗ ( x ) g ∗ ( x ) d x {\displaystyle \int _{\mathbb {R} ^{n}}f(x)g(x)\,dx\leq \int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx} where f ∗ {\displaystyle f^{*}} and g ∗ {\displaystyle g^{*}} are the symmetric decreasing rearrangements of f {\displaystyle f} and g {\displaystyle g} , respectively. The decreasing rearrangement f ∗ {\displaystyle f^{*}} of f {\displaystyle f} is defined via the property that for all r > 0 {\displaystyle r>0} the two super-level sets E f ( r ) = { x ∈ X : f ( x ) > r } {\displaystyle E_{f}(r)=\left\{x\in X:f(x)>r\right\}\quad } and E f ∗ ( r ) = { x ∈ X : f ∗ ( x ) > r } {\displaystyle \quad E_{f^{*}}(r)=\left\{x\in X:f^{*}(x)>r\right\}} have the same volume ( n {\displaystyle n} -dimensional Lebesgue measure) and E f ∗ ( r ) {\displaystyle E_{f^{*}}(r)} is a ball in R n {\displaystyle \mathbb {R} ^{n}} centered at x = 0 {\displaystyle x=0} , i.e.

Source: Wikipedia "Hardy–Littlewood inequality" · CC BY-SA 4.0

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