section10 notes

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

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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...
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