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 < +∞,

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

For 1 < p < 2,

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

where

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

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

$x\mapsto x^p.$

References

• "Uniformly convex spaces", Transactions of the American Mathematical Society 40 (3): 396–414, 1936, doi:10.2307/1989630 .
• "On the uniform convexity of L^p and ℓ^p", Arkiv för Matematik 3 (3): 239–244, 1956, doi:10.1007/BF02589410, Bibcode: 1956ArM.....3..239H .
• "On Clarkson's inequalities", Communications on Pure and Applied Mathematics 23 (4): 603–607, 1970, doi:10.1002/cpa.3160230405 .

External links

• Clarkson inequality at PlanetMath.org.