Clarkson's inequalities - Alchetron, the free social encyclopedia

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 < +∞,

∥ f + g 2 ∥ L p p + ∥ f − g 2 ∥ L p p ≤ 1 2 ( ∥ f ∥ L p p + ∥ g ∥ L p p ) .

For 1 < p < 2,

∥ f + g 2 ∥ L p q + ∥ f − g 2 ∥ L p q ≤ ( 1 2 ∥ f ∥ L p p + 1 2 ∥ g ∥ L p p ) 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 ↦ x p .

References

Clarkson's inequalities Wikipedia

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