section10 notes

# Rn this is all for u s rn but s rn h m rn is dense

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: n Rn Now, if m > n/2 then the second integral is ﬁnite. Since the ﬁrst integral is the norm on H m (Rn ) we see that (10.2) sup |u(x)| = ∈u∈L� ∀ (2� )−n ∈u∈H m , m > n/2 . Rn This is all for u → S (Rn ), but S (Rn ) γ� H m (Rn ) is dense. The estimate (10.2) shows that if uj � u in H m (Rn ), with uj → S (Rn ), 0 then uj � u� in C0 (Rn ). In fact u� = u in S � (Rn ) since uj � u in 0 L2 (Rn ) and uj � u� in C0 (Rn ) both imply that uj α converges, so � � � uj α � uα = u� α � α → S (Rn ). Rn Rn Rn � Notice here the precise meaning of u = u� , u → H m (Rn ) � L2 (Rn ), 0 u → C0 (Rn ). When identifying u → L2 (Rn ) with the corresponding tempered distribution, the values on any set of measure zero ‘are lost’. Thus as functions (10.1) means that each u → H m (Rn ) has a represen­ 0 tative u� → C0 (Rn ). We can extend this to higher derivatives by noting that � 64 RICHARD B. MELROSE Proposition 10.2. If u → H m (Rn...
View Full Document

Ask a homework question - tutors are online