Unformatted text preview: Sobolev embedding
theorem we deduce that, with m > k + n/2,
u(α) ∀ C ∈⊂x≤k α∈H m � α → S (Rn ).
This is the same as
� −k
�
�⊂x≤ u(α)� ∀ C ∈α∈H m � α → S (Rn ). which shows that ⊂x≤−k u → H −m (Rn ), i.e., from Proposition 9.8,
�
⊂x≤−k u =
D � u� , u� → L2 (Rn ) .
��m In fact, choose j > n/2 and consider v� → H j (Rn ) deﬁned by v� =
ˆ
⊂π ≤−j u� . As in the proof of Proposition 9.14 we conclude that
ˆ
�
�
�
0
u� =
D � u�,� , u�,� → H j (Rn ) � C0 (Rn ) .
� �j Thus,17 (10.10) u = ⊂x≤k � �
0
D� v� , v� → C0 (Rn ) . � �M To get (10.9) we ‘commute’ the factor ⊂x≤k to the inside; since I have
not done such an argument carefully so far, let me do it as a lemma.
17This is probably the most useful form of the representation theorem! 66 RICHARD B. MELROSE Lemma 10.6. For any � → Nn there are polynomials p�,� (x) of degrees
0
at most � − � such that
�
�
�
⊂x≤k D � v =
D � −� p�,� ⊂x≤k−2� −� v .
��� Proof. In fact it is convenient to prove a more general result. Suppose
p is a p olynomial of a degree at most j then there exist polynomials of
degrees at most j + � − � such that...
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 Fall '13
 Melrose
 Derivative

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