We have Op w (a)u ? S (R n ) and ,
?(1, g) For any ?, r ? R, with Proposition 2.5, the operator H maps H ?,r (R n ) into itself continuously and is invertible, for ? sufficiently large, arguing as in the proof of Lemma 2.3. We thus set?uset? set?u = H ?1 u ? H ?,r (R n ) ,
R n ) the injectivity of Op w (? ? ?s µ ?k ) from L 2 into H k,s (R n ) (see Lemma 2.3) we obtain Op w (? ? s µ k )? u = v. In particular, as ?, r ? R are chosen arbitrarily we find v ? H k ? ,s ? (R n ) for any k ? , s ? ? R, if ? is chosen sufficiently large and we have by (B.17) v k ? ,s ? = Op w (? ? s ? µ k ? ) ,
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