Clarkson's inequalities

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

Statement of the inequalities[edit]

Let (X, Σ, μ) be a measure space; let fg : X → R be measurable functions in Lp. 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. \,

References[edit]

External links[edit]