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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+g2‖Lpp+‖f−g2‖Lpp≤12(‖f‖Lpp+‖g‖Lpp)._L^p^p+left

For 1 < p < 2,

‖f+g2‖Lpq+‖f−g2‖Lpq≤(12‖f‖Lpp+12‖g‖Lpp)qp,_L^p^qleq left(frac 12

where

1p+1q=1,displaystyle frac 1p+frac 1q=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↦xp.displaystyle xmapsto 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.