# hw5 - Math 245C Homework 5 Brett Hemenway June 4 2007...

This preview shows pages 1–3. 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: Math 245C Homework 5 Brett Hemenway June 4, 2007 Folland Chapter 9 29. (a) Suppose F = μ ∈ M ( R n ). Then by proposition 8.49, k μ * φ k p ≤ k φ k p k μ k , so F ∈ C 1 . On the other hand, if F ∈ C 1 , then let { φ n } be an approximate identity. Then sup n k F * φ n k p ≤ sup n C k φ n k p < ∞ . Thus the family { F * φ n } lies in a closed ball in C ( R n ) * . By Alaoglu’s Theorem, this ball is compact, and so the sequence { F * φ n } has a subsequence converging to μ ∈ C ( R n ) * , i.e. lim k →∞ h F * φ n k ,g i = h μ,g i∀ g ∈ C ( R n ) . It remains to show that F = μ as distributions. Since { φ n k } is an approximate identity, φ n k * g → g . Thus h μ,g i = lim k →∞ h F * φ n k ,g i = lim k →∞ h F,φ n k * g i = h F,g i . Thus F ∈ C ( R n ) * = M ( R n ). (b) Suppose F ∈ S , and ˆ F ∈ L ∞ . Then applying Plancherel’s The- orem twice, we have k F * φ k 2 = k [ F * φ k 2 = k ˆ F ˆ φ k 2 ≤ k ˆ F k ∞ k ˆ φ k 2 = k ˆ F k ∞ k φ k 2 . 1 Thus F ∈ C 2 . Conversely, assume F ∈ C 2 . Let g ∈ C ∞ c ( R n ). Then k g ˆ F k 2 = k [ ˇ g * F k 2 = k ˇ g * F k 2 ≤ C k ˇ g k 2 = C k g k 2 . Thus ≤ C 2 k g k 2 2- k g ˆ F k 2 2 = Z ( C 2- ˆ F 2 ) g 2 , for all g ∈ C ∞ c . Thus C 2- ˆ F 2 ≥ 0 almost everywhere. Which means k ˆ F k ∞ ≤ C...
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

hw5 - Math 245C Homework 5 Brett Hemenway June 4 2007...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online