Prob 6 - ans 1

Prob 6 - ans 1 - 1 H¨older’s Inequality and...

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Unformatted text preview: 1. H¨older’s Inequality and Minkowski’s inequality. We fix p, q ∈ [1 , ∞ ] such that 1 p + 1 q = 1 . Definition 1.1. Suppose f : R n → R is Lebesgue measurable. We let || f || p = Z | f ( x ) | p dx ¶ 1 /p if p < ∞ and we let || f || ∞ = sup { t ∈ (0 , ∞ ) : L n ( {| f | > t } ) > } . Proposition 1.1. Suppose f : R n → R is Lebesgue measurable and c ∈ R . Then || cf || p = | c ||| f || p . Proof. Exercise for the reader. / Theorem 1.1. (H¨older’s Inequality.) Suppose f, g : R n → R are Lebesgue measurable. Then || fg || 1 ≤ || f || p || g || q . Proof. Exercise for the reader. Here are some hints. Treat the case p = ∞ or q = ∞ by a straightforward argument. When p < ∞ and q < ∞ first reduce to the case || f || p = 1 and || g || q = 1 by making use of the previous Proposition; then apply the inequality a 1 /p b 1 /q ≤ 1 p a + 1 q b for a, b ∈ (0 , ∞ ). / Theorem 1.2. Minkowski’s Inequality. Suppose f, g : R n → R are Lebesgue measurable. Then || f + g || p ≤ || f || p + || g || p . Proof. Exercise for the reader. Here are some hints. The cases p = 1 and p = ∞ follow from the triangle inequality. In case 1 < p < ∞ apply H¨older’s Inequality to | f + g | p ≤ | f + g | p- 1 ( | f | + | g | ). / 1.1. An extension of H¨older’s Inequality. Suppose p, q, r ∈ [0 , ∞ ] and 1 p + 1 q = 1 r . Theorem 1.3. Suppose f, g : R n → R are Lebesgue measurable. Then || fg || r ≤ || f || p || g || q ....
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This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.

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Prob 6 - ans 1 - 1 H¨older’s Inequality and...

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