Clarkson's inequalities explained

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 < +∞,

\left\| (f + g)/(2) \right\| ^{p} _{L ^{p}} + \left\| (f - g)/(2) \right\| ^{p} _{L ^{p}} \le (1)/(2) \left( \| f \| ^{p} _{L ^{p}} + \| g \| ^{p} _{L ^{p}} \right).

For 1 < p < 2,

\left\| (f + g)/(2) \right\| ^{q} _{L ^{p}} + \left\| (f - g)/(2) \right\| ^{q} _{L ^{p}} \le \left( (1)/(2) \| f \| ^{p} _{L ^{p}} + (1)/(2) \| g \| ^{p} _{L ^{p}} \right) ^{(q)/(p)} ,

where

(1 {p})/(+) (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

x \mapsto x ^{p.}

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