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17this is probably the most useful form of the

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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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