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 o
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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 \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). For 1 \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 , where : \frac1 p + \frac1 q = 1, i.e., q = p ⁄ (p − 1).
