Prob 6 - ans 1

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

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.

### Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online