Clarkson's inequalities

In mathematics, Clarkson's inequalities 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 < +∞,

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

For 1 < p < 2,

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

where

$\frac1{p} + \frac1{q} = 1,$

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

$x \mapsto x^{p}.$

External links

• Clarkson inequality at PlanetMath.