This preview shows page 1. Sign up to view the full content.
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
 Fall '13
 Melrose
 Derivative

Click to edit the document details