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 < +∞,

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

For 1 < p < 2,

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

where

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

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

${\displaystyle x\mapsto x^{p}.\,}$

References

• Clarkson, James A. (1936), "Uniformly convex spaces", Transactions of the American Mathematical Society, 40 (3): 396–414, doi:10.2307/1989630, MR 1501880.
• Hanner, Olof (1956), "On the uniform convexity of L^p and ℓ^p", Arkiv för Matematik, 3 (3): 239–244, doi:10.1007/BF02589410, MR 0077087.
• Friedrichs, K. O. (1970), "On Clarkson's inequalities", Communications on Pure and Applied Mathematics, 23: 603–607, doi:10.1002/cpa.3160230405, MR 0264372.

External links

• Clarkson inequality at PlanetMath.org.