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 < +∞,

[math]\displaystyle{ \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). }[/math]

For 1 < p < 2,

[math]\displaystyle{ \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}, }[/math]

where

[math]\displaystyle{ \frac1{p} + \frac1{q} = 1, }[/math]

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

[math]\displaystyle{ x \mapsto x^p. }[/math]

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.