Clarkson's inequalities

In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of L^p spaces. They give bounds for the L^p-norms of the sum and difference of two measurable functions in L^p in terms of the L^p-norms of those functions individually.

Statement of the inequalities

Let (X, Σ, μ) be a measure space; let f, g : X → R be measurable functions in L^p. Then, for 2 ≤ p < +∞,

${\displaystyle \left\|{\frac {f+g}{2}}\right\|_{L^{p}}^{p}+\left\|{\frac {f-g}{2}}\right\|_{L^{p}}^{p}\leq {\frac {1}{2}}\left(\|f\|_{L^{p}}^{p}+\|g\|_{L^{p}}^{p}\right).}$

For 1 < p < 2,

${\displaystyle \left\|{\frac {f+g}{2}}\right\|_{L^{p}}^{q}+\left\|{\frac {f-g}{2}}\right\|_{L^{p}}^{q}\leq \left({\frac {1}{2}}\|f\|_{L^{p}}^{p}+{\frac {1}{2}}\|g\|_{L^{p}}^{p}\right)^{\frac {q}{p}},}$

where

${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1,}$

i.e., q = p ⁄ (p − 1).

The case p ≥ 2 is somewhat easier to prove, being a simple application of the triangle inequality and the convexity of

${\displaystyle x\mapsto x^{p}.}$

References

• MR 1501880
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• MR 0077087
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• MR 0264372
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• Clarkson inequality at PlanetMath.